Geometry and Trigonometry Practice Questions — JAMB UTME — Page 4

Question 61Question →

The sum of the interior angles of a regular polygon is $1440^\circ$ . What is the measure, in degrees, of one exterior angle of this polygon?

Show answer & explanation

Answer: 36

Answer

The measure of one exterior angle of the polygon is

36^\circ

.

The sum of the interior angles of an

n

-sided polygon is given by

(n-2) \times 180^\circ

. Setting

(n-2) \times 180^\circ = 1440^\circ

gives

n-2 = 8

, so the polygon has

n = 10

sides (a decagon). The measure of each exterior angle of a regular polygon is

\frac{360^\circ}{n} = \frac{360^\circ}{10} = 36^\circ

.

Step-by-Step Solution

1

Set up the equation for the sum of interior angles of an

n

-sided polygon.

(n - 2) \times 180^\circ = 1440^\circ

The sum of interior angles of any convex

n

-sided polygon is

(n - 2) \times 180^\circ

.

2

Solve for

n

, the number of sides.

n - 2 = \frac{1440}{180} = 8 \implies n = 10

Dividing the interior angle sum by

180^\circ

gives

n - 2

.

3

Calculate the measure of one exterior angle.

Exterior angle

= \frac{360^\circ}{10} = 36^\circ

The sum of exterior angles of any convex polygon is

360^\circ

, so each exterior angle of a regular polygon with

n

sides is

\frac{360^\circ}{n}

.

Key Concept

Relationship between interior angle sum, number of sides, and exterior angles of a regular polygon.

Question 62Question →

The interior angles of a convex pentagon are given as $(2x + 10)^\circ$ , $(x + 25)^\circ$ , $(3x - 15)^\circ$ , $(2x + 30)^\circ$ , and $(2x + 10)^\circ$ . What is the value of $x$ ?

30

48

60

66

Show answer & explanation

Answer: 48

Answer

The value of

x

is 48.

For a pentagon (

n = 5

), the sum of interior angles is

(5 - 2) \times 180^\circ = 540^\circ

. Summing the five given interior angle expressions yields

10x + 60

. Equating

10x + 60 = 540

gives

10x = 480

, which solves to

x = 48

.

Step-by-Step Solution

1

Calculate the sum of interior angles of a 5-sided polygon (pentagon)

Sum

= (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ

The formula for the sum of interior angles of a polygon with

n

sides is

(n - 2) \times 180^\circ

.

2

Sum the given algebraic expressions for the interior angles

(2x + 10) + (x + 25) + (3x - 15) + (2x + 30) + (2x + 10) = 10x + 60

Combine like terms for the

x

terms and constant terms.

3

Equate the sum of expressions to

540^\circ

and solve for

x

10x + 60 = 540 \implies 10x = 480 \implies x = 48

Subtract 60 from both sides and divide by 10 to isolate

x

.

Key Concept

Interior Angle Sum of Polygons

Estimated Time:1m 15s

Question 63Question →

In a regular polygon, the measure of each interior angle is $132^\circ$ greater than the measure of each exterior angle. How many sides does this polygon have?

Show answer & explanation

Answer: 15

Answer

The polygon has 15 sides.

Since the interior angle

I

and exterior angle

E

of a regular polygon sum to

180^\circ

(

I + E = 180^\circ

) and their given difference is

I - E = 132^\circ

, subtracting the difference equation from the sum equation gives

2E = 48^\circ

, which simplifies to

E = 24^\circ

. The number of sides is

n = \frac{360^\circ}{E} = \frac{360^\circ}{24^\circ} = 15

.

Step-by-Step Solution

1

Set up the linear pair equation for interior and exterior angles

I + E = 180^\circ

An interior angle and its adjacent exterior angle at any vertex of a polygon lie on a straight line and sum to

180^\circ

.

2

Set up the given condition equation

I - E = 132^\circ

The question states that each interior angle is

132^\circ

greater than each exterior angle.

3

Solve for the exterior angle

E

E = 24^\circ

Subtracting

I - E = 132^\circ

from

I + E = 180^\circ

yields

2E = 48^\circ

, giving

E = 24^\circ

.

4

Calculate the number of sides

n

n = 15

The sum of all exterior angles of any convex polygon is

360^\circ

, so

n = \frac{360^\circ}{E} = \frac{360^\circ}{24^\circ} = 15

.

Key Concept

Interior and Exterior Angle Properties of Regular Polygons

Question 64Question →

Determine the equation of the locus of a point $P(x, y)$ that is equidistant from the two parallel lines $2x - 3y + 6 = 0$ and $2x - 3y - 4 = 0$ .

Show answer & explanation

Answer: 2x - 3y + 1 = 0; 2x-3y+1=0; 2x - 3y = -1; 2x-3y=-1

Answer

2x - 3y + 1 = 0

The locus of points equidistant from two parallel lines

ax + by + c_1 = 0

and

ax + by + c_2 = 0

is a parallel line midway between them, defined by

ax + by + \frac{c_1 + c_2}{2} = 0

. Substituting

c_1 = 6

and

c_2 = -4

yields

\frac{6 + (-4)}{2} = 1

, giving the equation

2x - 3y + 1 = 0

.

Step-by-Step Solution

1

Identify the geometric principle for the locus between two parallel lines.

The locus of points equidistant from two parallel lines

ax + by + c_1 = 0

and

ax + by + c_2 = 0

is a third parallel line given by

ax + by + \frac{c_1 + c_2}{2} = 0

.

Points equidistant from two parallel lines lie on a parallel line midway between them.

2

Calculate the average of the constant terms

c_1 = 6

and

c_2 = -4

.

c_{mid} = \frac{6 + (-4)}{2} = \frac{2}{2} = 1

.

The midpoint constant term is the arithmetic mean of the two original constants.

3

Construct the equation of the locus line.

2x - 3y + 1 = 0

.

Combining the common linear coefficients

2x - 3y

with the calculated midpoint constant

1

gives the required equation.

Key Concept

Locus equidistant from two parallel lines

Estimated Time:1m 30s

Question 65Question →

An open storage container is designed in the shape of a frustum of a right circular cone. The top radius of the container is $4\text{ m}$ , the bottom radius is $1\text{ m}$ , and its vertical height is $7\text{ m}$ . Taking $\pi = \frac{22}{7}$ , what is the volume of the container in cubic metres ( $\text{m}^3$ )?

Show answer & explanation

Answer: 154

Answer

The volume of the container is

154\text{ m}^3

.

Using the volume formula for a frustum of a cone

V = \frac{1}{3}\pi h (R^2 + Rr + r^2)

with

R = 4\text{ m}

,

r = 1\text{ m}

,

h = 7\text{ m}

, and

\pi = \frac{22}{7}

yields

V = \frac{1}{3} \times \frac{22}{7} \times 7 \times (16 + 4 + 1) = 22 \times 7 = 154\text{ m}^3

.

Step-by-Step Solution

1

Identify the given dimensions of the frustum of the cone.

Top radius

R = 4\text{ m}

, bottom radius

r = 1\text{ m}

, height

h = 7\text{ m}

, and

\pi = \frac{22}{7}

.

Establishing known parameter values for the frustum volume formula.

2

Apply the standard formula for the volume of a frustum of a circular cone.

V = \frac{1}{3}\pi h (R^2 + Rr + r^2)

This formula accounts for the non-uniform cross-sectional area of a truncated cone.

3

Calculate the sum of the squares and the product of the radii.

R^2 + Rr + r^2 = 4^2 + (4)(1) + 1^2 = 16 + 4 + 1 = 21

Evaluating the quadratic radius factor in the frustum equation.

4

Substitute all numeric values and simplify.

V = \frac{1}{3} \times \frac{22}{7} \times 7 \times 21 = 22 \times 7 = 154\text{ m}^3

Simplifying by cancelling

7

in the numerator and denominator, then dividing

21

by

3

yields

22 \times 7 = 154

.

Key Concept

Volume of a Frustum of a Cone

Question 66Question →

A point $P(x, y)$ moves in a plane such that the line segment joining the fixed points $A(1, 2)$ and $B(5, 6)$ subtends a right angle at $P$ . Which of the following equations represents the locus of $P$ ?

x^2 + y^2 - 6x - 8y + 17 = 0

x - y + 1 = 0

x^2 + y^2 - 6x - 8y - 7 = 0

x^2 + y^2 + 6x + 8y + 17 = 0

Show answer & explanation

Answer: $x^2 + y^2 - 6x - 8y + 17 = 0$

Answer

The equation of the locus of

P

is

x^2 + y^2 - 6x - 8y + 17 = 0

.

The locus of a point

P

that subtends a

90^\circ

angle at two fixed points

A(1, 2)

and

B(5, 6)

forms a circle with

AB

as diameter. Using the gradient condition for perpendicular lines,

\frac{y-2}{x-1} \times \frac{y-6}{x-5} = -1

, which simplifies to

x^2 + y^2 - 6x - 8y + 17 = 0

.

Step-by-Step Solution

1

Apply the perpendicularity condition for the line segments

AP

and

BP

.

Gradient of

AP = \frac{y - 2}{x - 1}

and gradient of

BP = \frac{y - 6}{x - 5}

. Since

\angle APB = 90^\circ

, their product must be

-1

:

\left(\frac{y - 2}{x - 1}\right) \cdot \left(\frac{y - 6}{x - 5}\right) = -1

.

Two perpendicular line segments have gradients whose product is

-1

.

2

Multiply out the denominators and numerators.

(y - 2)(y - 6) = -(x - 1)(x - 5) \implies y^2 - 8y + 12 = -(x^2 - 6x + 5).

Algebraic expansion of the equation obtained from the gradient product.

3

Rearrange all terms to one side to express in standard second-degree form.

x^2 + y^2 - 6x - 8y + 17 = 0.

Rearranging yields the Cartesian equation of the locus.

Key Concept

Locus of a point subtending a right angle at two fixed points

Estimated Time:1m 30s

Question 67Question →

An isosceles trapezium has parallel sides of lengths $14\text{ cm}$ and $8\text{ cm}$ . If each of the non-parallel sides has a length of $5\text{ cm}$ , calculate the area of the trapezium in $\text{cm}^2$ .

Show answer & explanation

Answer: 44

Answer

The area of the trapezium is

44\text{ cm}^2

.

Projecting the top base of length

8\text{ cm}

onto the bottom base of length

14\text{ cm}

leaves a difference of

6\text{ cm}

, which is divided equally into two

3\text{ cm}

segments on either side. Using the Pythagorean theorem with the non-parallel side (

5\text{ cm}

) and the base segment (

3\text{ cm}

) gives a height of

4\text{ cm}

. The area is then calculated as

\frac{1}{2} \times (14 + 8) \times 4 = 44\text{ cm}^2

.

Step-by-Step Solution

1

Determine the projection segment length on the longer base

x = \frac{14 - 8}{2} = 3\text{ cm}

Since the trapezium is isosceles, dropping perpendiculars from both ends of the top base creates two identical right-angled triangles at the sides.

2

Calculate the perpendicular height using the Pythagorean theorem

h = \sqrt{5^2 - 3^2} = \sqrt{16} = 4\text{ cm}

The slant side (

5\text{ cm}

), the height (

h

), and the projection segment (

3\text{ cm}

) form a right-angled triangle.

3

Calculate the area of the trapezium

\text{Area} = \frac{1}{2}(14 + 8) \times 4 = 44\text{ cm}^2

The area of a trapezium is given by half the sum of its parallel sides multiplied by its perpendicular height.

Key Concept

Perimeter and Area of Plane Shapes - Area of Isosceles Trapezium

Question 68Question →

Given that $\theta$ is an acute angle satisfying $\sec \theta + \tan \theta = 3$ , what is the exact numerical value of $10 \sin \theta$ ?

Show answer & explanation

Answer: 8

Answer

The numerical value of

10 \sin \theta

is 8.

Using the standard fundamental identity

\sec^2 \theta - \tan^2 \theta = 1

, we factor it into

(\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = 1

. Substituting the given value

\sec \theta + \tan \theta = 3

gives

\sec \theta - \tan \theta = \frac{1}{3}

. Solving these linear equations yields

\sec \theta = \frac{5}{3}

(so

\cos \theta = \frac{3}{5}

) and

\tan \theta = \frac{4}{3}

. Using

\sin \theta = \tan \theta \cdot \cos \theta

, we find

\sin \theta = \frac{4}{5} = 0.8

. Multiplying by 10 gives the exact result 8.

Step-by-Step Solution

1

Apply the trigonometric identity

\sec^2 \theta - \tan^2 \theta = 1

(\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = 1

Difference of squares factorization links

\sec \theta + \tan \theta

and

\sec \theta - \tan \theta

as reciprocals.

2

Substitute

\sec \theta + \tan \theta = 3

to find

\sec \theta - \tan \theta

\sec \theta - \tan \theta = \frac{1}{3}

Dividing both sides of

3(\sec \theta - \tan \theta) = 1

by 3.

3

Solve the system of equations for

\sec \theta

and

\tan \theta

\sec \theta = \frac{5}{3} \text{ and } \tan \theta = \frac{4}{3}

Adding equations gives

2\sec \theta = \frac{10}{3}

; subtracting gives

2\tan \theta = \frac{8}{3}

.

4

Calculate

\sin \theta

and evaluate

10 \sin \theta

\sin \theta = \frac{4}{5} \implies 10 \sin \theta = 8

Since

\cos \theta = \frac{3}{5}

and

\tan \theta = \frac{4}{3}

,

\sin \theta = \tan \theta \cdot \cos \theta = \frac{4}{5}

.

Key Concept

Trigonometric identities relating secant and tangent ratios

Question 69Question →

Which set contains all the solutions to the trigonometric equation $\sin 2x = \cos x$ for $0^\circ \le x \le 180^\circ$ ?

\{30^\circ, 90^\circ, 150^\circ\}

\{30^\circ, 150^\circ\}

\{30^\circ, 90^\circ\}

\{60^\circ, 90^\circ, 120^\circ\}

Show answer & explanation

Answer: $\{30^\circ, 90^\circ, 150^\circ\}$

Answer

The correct set of solutions is \{30^\circ, 90^\circ, 150^\circ\}.

Using the identity

\sin 2x = 2\sin x \cos x

, the equation becomes

2\sin x \cos x - \cos x = 0

. Factoring out

\cos x

gives

\cos x(2\sin x - 1) = 0

. Setting each factor to zero yields

\cos x = 0

(giving

x = 90^\circ

) and

\sin x = \frac{1}{2}

(giving

x = 30^\circ

and

x = 150^\circ

). Thus, the complete set of solutions in the given interval is \{30^\circ, 90^\circ, 150^\circ\}.

Step-by-Step Solution

1

Apply the double-angle identity for sine.

Substitute

\sin 2x = 2\sin x \cos x

into the equation to get

2\sin x \cos x = \cos x

.

This expresses the equation in terms of single angle

x

.

2

Rearrange and factor the equation.

2\sin x \cos x - \cos x = 0 \implies \cos x(2\sin x - 1) = 0

.

Factoring prevents losing valid roots that occur when a variable factor equals zero.

3

Set each factor to zero and solve for

x

in the interval

0^\circ \le x \le 180^\circ

.

First factor:

\cos x = 0 \implies x = 90^\circ

.
Second factor:

2\sin x - 1 = 0 \implies \sin x = \frac{1}{2} \implies x = 30^\circ

or

x = 180^\circ - 30^\circ = 150^\circ

.

Finding all principal and secondary angles within the specified domain.

4

Combine all valid solutions into a set.

x \in \{30^\circ, 90^\circ, 150^\circ\}

.

All three values satisfy the original equation and lie within

0^\circ \le x \le 180^\circ

.

Key Concept

Solving trigonometric equations using identities and factoring

Estimated Time:1m 30s

Question 70Question →

Which of the following sets contains all values of $x$ in the interval $0^\circ \le x \le 360^\circ$ that satisfy the trigonometric equation $\sqrt{3}\sin x + \cos x = 0$ ?

150^\circ \text{ and } 330^\circ

30^\circ \text{ and } 210^\circ

120^\circ \text{ and } 300^\circ

150^\circ \text{ only}

Show answer & explanation

Answer: $150^\circ \text{ and } 330^\circ$

Answer

150^\circ \text{ and } 330^\circ

The given equation

\sqrt{3}\sin x + \cos x = 0

simplifies to

\tan x = -\frac{1}{\sqrt{3}}

. Since tangent is negative in the second and fourth quadrants with a reference angle of

30^\circ

, the solutions in the domain

0^\circ \le x \le 360^\circ

are

180^\circ - 30^\circ = 150^\circ

and

360^\circ - 30^\circ = 330^\circ

.

Step-by-Step Solution

1

Rearrange the trigonometric equation into single ratio form

\sqrt{3}\sin x = -\cos x \implies \frac{\sin x}{\cos x} = -\frac{1}{\sqrt{3}} \implies \tan x = -\frac{1}{\sqrt{3}}

Dividing both sides by

\cos x

converts the sum of sine and cosine terms into a simple tangent equation.

2

Determine the reference angle

\text{Reference angle } \alpha = 30^\circ

The acute angle whose tangent is

\frac{1}{\sqrt{3}}

is

30^\circ

.

3

Identify the quadrants and find all solutions in

0^\circ \le x \le 360^\circ

x = 180^\circ - 30^\circ = 150^\circ

(Quadrant II) and

x = 360^\circ - 30^\circ = 330^\circ

(Quadrant IV)

The tangent function is negative in Quadrants II and IV.

Key Concept

Solving simple trigonometric equations by reducing to a basic ratio and finding all solutions within a given domain.

Estimated Time:1m 30s

Question 71Question →

In triangle $XYZ$ , side length $x = 3\text{ cm}$ , side length $y = 8\text{ cm}$ , and the included angle $\angle Z = 60^\circ$ . What is the length of side $z$ in cm?

Show answer & explanation

Answer: 7

Answer

The length of side

z

is

7\text{ cm}

.

Using the Cosine Rule

z^2 = x^2 + y^2 - 2xy \cos Z

, substituting

x = 3

,

y = 8

, and

\angle Z = 60^\circ

yields

z^2 = 3^2 + 8^2 - 2(3)(8)(0.5) = 9 + 64 - 24 = 49

. Taking the positive square root gives

z = 7\text{ cm}

.

Step-by-Step Solution

1

Identify known triangle components and select the appropriate rule

Two sides and the included angle (SAS) are given:

x = 3

,

y = 8

,

\angle Z = 60^\circ

, requiring the Cosine Rule.

When given two sides and the included angle (SAS), the Cosine Rule is used to find the third side.

2

Substitute values into the Cosine Rule formula

z^2 = x^2 + y^2 - 2xy \cos Z

z^2 = 3^2 + 8^2 - 2(3)(8)\cos 60^\circ

Direct algebraic substitution of side lengths and angle measure into the Cosine Rule.

3

Evaluate the trigonometric term and simplify the arithmetic expression

z^2 = 9 + 64 - 48(0.5) = 73 - 24 = 49

The exact value of

\cos 60^\circ

is

0.5

.

4

Solve for side length

z

by taking the principal square root

z = \sqrt{49} = 7\text{ cm}

Side length must be positive.

Key Concept

Applying the Cosine Rule to find the third side of a non-right-angled triangle given two sides and an included angle (SAS).

Estimated Time:1m 30s

Question 72Question →

A circle has a circumference of $44\text{ cm}$ . What is the area of the circle? (Take $\pi = \frac{22}{7}$ )

154\text{ cm}^2

308\text{ cm}^2

616\text{ cm}^2

44\text{ cm}^2

Show answer & explanation

Answer: $154\text{ cm}^2$

Answer

The area of the circle is

154\text{ cm}^2

.

First, find the radius from the circumference:

C = 2\pi r \implies 44 = 2 \times \frac{22}{7} \times r \implies r = 7\text{ cm}

. Then compute the area:

A = \pi r^2 = \frac{22}{7} \times 7^2 = 154\text{ cm}^2

. Therefore, the value

154\text{ cm}^2

is correct.

Step-by-Step Solution

1

Calculate the radius of the circle using the circumference formula

r = 7\text{ cm}

The formula for circumference is

C = 2\pi r

. Substituting

C = 44

and

\pi = \frac{22}{7}

yields

44 = 2 \times \frac{22}{7} \times r

, which simplifies to

\frac{44}{7}r = 44

, so

r = 7\text{ cm}

.

2

Calculate the area using the area formula for a circle

Area =

154\text{ cm}^2

The area formula is

A = \pi r^2

. Substituting

r = 7

and

\pi = \frac{22}{7}

gives

A = \frac{22}{7} \times 7^2 = 22 \times 7 = 154\text{ cm}^2

.

Key Concept

Relationship between circumference and area of a circle

Question 73Question →

Two regular polygons, $P_1$ and $P_2$ , have $n$ sides and $2n$ sides respectively. If the sum of one interior angle of $P_1$ and one exterior angle of $P_2$ is $150^\circ$ , what is the total number of diagonals of polygon $P_2$ ?

9

54

66

108

Show answer & explanation

Answer: 54

Answer

The total number of diagonals of polygon

P_2

is 54.

The interior angle of an

n

-sided polygon is

180^\circ - \frac{360^\circ}{n}

and the exterior angle of a

2n

-sided polygon is

\frac{180^\circ}{n}

. Adding these gives

180^\circ - \frac{180^\circ}{n} = 150^\circ

, which yields

n = 6

. Polygon

P_2

therefore has

12

sides, and the number of diagonals is

\frac{12 \times 9}{2} = 54

.

Step-by-Step Solution

1

Express the interior angle of

P_1

and the exterior angle of

P_2

in terms of

n

.

Interior angle of

P_1 = 180^\circ - \frac{360^\circ}{n}

; Exterior angle of

P_2 = \frac{360^\circ}{2n} = \frac{180^\circ}{n}

.

The interior angle of an

n

-sided regular polygon is

180^\circ - \frac{360^\circ}{n}

, and the exterior angle of a

2n

-sided regular polygon is

\frac{360^\circ}{2n}

.

2

Set up and solve the angle sum equation.

\left(180^\circ - \frac{360^\circ}{n}\right) + \frac{180^\circ}{n} = 150^\circ \implies 180^\circ - \frac{180^\circ}{n} = 150^\circ \implies \frac{180^\circ}{n} = 30^\circ \implies n = 6$.

The sum of the two angles is given as

150^\circ

.

3

Determine the number of sides of polygon

P_2

.

Polygon

P_2

has

2n = 2(6) = 12

sides.

Polygon

P_2

has

2n

sides.

4

Calculate the total number of diagonals for a polygon with 12 sides using

D = \frac{k(k-3)}{2}

.

D = \frac{12(12 - 3)}{2} = \frac{12 \times 9}{2} = 54.

The formula for the number of diagonals in a polygon with

k

sides is

\frac{k(k-3)}{2}

.

Key Concept

Interior and exterior angles of regular polygons and the polygon diagonal count formula

Question 74Question →

A composite plane figure is formed by joining a major sector of a circle of radius $14\text{ cm}$ (having a central angle of $270^\circ$ and center $O$ ) to a square $OACB$ . The vertices $A$ and $B$ of the square lie on the circle such that $OA$ and $OB$ serve as the bounding radii of the major sector. What is the total area of the combined figure in $\text{cm}^2$ ? (Take $\pi = \frac{22}{7}$ )

350\text{ cm}^2

504\text{ cm}^2

560\text{ cm}^2

658\text{ cm}^2

Show answer & explanation

Answer: $658\text{ cm}^2$

Answer

658\text{ cm}^2

The total area of the composite figure is the sum of the non-overlapping regions: the major sector of central angle

270^\circ

(

462\text{ cm}^2

) and the square of side length

14\text{ cm}

(

196\text{ cm}^2

), giving

462 + 196 = 658\text{ cm}^2

.

Step-by-Step Solution

1

Calculate the area of the major sector of central angle

270^\circ

and radius

14\text{ cm}

.

\text{Area}_{\text{sector}} = \frac{270^\circ}{360^\circ} \times \pi r^2 = \frac{3}{4} \times \frac{22}{7} \times 14^2 = \frac{3}{4} \times 616 = 462\text{ cm}^2

The major sector covers

\frac{270}{360} = \frac{3}{4}

of the full circle.

2

Calculate the area of the square

OACB

with side length

s = OA = 14\text{ cm}

.

\text{Area}_{\text{square}} = s^2 = 14^2 = 196\text{ cm}^2

The radii

OA

and

OB

form two adjacent sides of the square

OACB

of length

14\text{ cm}

.

3

Sum the areas of the major sector and the square to find the total composite area.

\text{Total Area} = 462\text{ cm}^2 + 196\text{ cm}^2 = 658\text{ cm}^2

The major sector and square share boundaries

OA

and

OB

without interior overlap.

Key Concept

Perimeter and Area of Plane Shapes

Estimated Time:2m 0s

Question 75Question →

Find the equation of the locus of a point $P(x, y)$ that moves such that its distance from the fixed point $(3, 0)$ is always equal to its perpendicular distance from the vertical line $x = -3$ .

Show answer & explanation

Answer: y^2 = 12x; y^2 - 12x = 0; y^2=12x; y^2 - 12x = 0

Answer

The equation of the locus is

y^2 = 12x

(or

y^2 - 12x = 0

).

Equating the distance from

P(x, y)

to

(3, 0)

, which is

\sqrt{(x - 3)^2 + y^2}

, and the distance from

P(x, y)

to

x = -3

, which is

|x + 3|

, squaring both sides gives

x^2 - 6x + 9 + y^2 = x^2 + 6x + 9

. Subtracting

x^2 + 9

from both sides yields

y^2 = 12x

.

Step-by-Step Solution

1

Express the distance from

P(x, y)

to the point

(3, 0)

using the distance formula.

d_1 = \sqrt{(x - 3)^2 + (y - 0)^2} = \sqrt{(x - 3)^2 + y^2}

The distance between two points

(x_1, y_1)

and

(x_2, y_2)

is given by

\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

.

2

Express the perpendicular distance from

P(x, y)

to the line

x = -3

.

d_2 = |x - (-3)| = |x + 3|

The perpendicular distance from a point

(x, y)

to a vertical line

x = k

is given by

|x - k|

.

3

Set the two distance expressions equal according to the locus condition.

\sqrt{(x - 3)^2 + y^2} = |x + 3|

The locus condition states that the distance to

(3, 0)

is equal to the distance to line

x = -3

.

4

Square both sides and simplify to obtain the Cartesian equation.

(x - 3)^2 + y^2 = (x + 3)^2 \implies x^2 - 6x + 9 + y^2 = x^2 + 6x + 9 \implies y^2 = 12x

Squaring eliminates the square root and absolute value signs, leading to the algebraic representation of the locus.

Key Concept

Definition and equation of a parabola as the locus of a point equidistant from a fixed point (focus) and a fixed line (directrix)

Alternative Method

Recognize that the definition of a parabola is the locus of points equidistant from a focus

(a, 0)

and a directrix

x = -a

. Here

a = 3

, so the standard equation

y^2 = 4ax

directly gives

y^2 = 4(3)x = 12x

.

Estimated Time:1m 30s

Question 76Question →

A sector of a circle of radius $6\text{ cm}$ has a total perimeter whose numerical value is equal to the numerical value of its area. What is the area of the sector, in $\text{cm}^2$ ?

Show answer & explanation

Answer: 18

Answer

The area of the sector is

18\text{ cm}^2

.

For a sector of radius

r = 6\text{ cm}

, its total perimeter is

P = 2(6) + s = 12 + s

, where

s

is the arc length. Its area is

A = \frac{1}{2}(6)s = 3s

. Setting

P = A

gives

12 + s = 3s \implies 2s = 12 \implies s = 6\text{ cm}

. Substituting

s = 6

into the area expression gives

A = 3(6) = 18\text{ cm}^2

.

Step-by-Step Solution

1

Express the perimeter and area of the sector in terms of the arc length

s

P = 2r + s = 12 + s

and

A = \frac{1}{2}rs = 3s

The total perimeter of a sector includes two straight radii plus the arc length, while its area is given by

\frac{1}{2}rs

.

2

Equate the numerical values of perimeter and area

12 + s = 3s

The question specifies that the numerical value of the total perimeter equals the numerical value of its area.

3

Solve for the arc length

s

2s = 12 \implies s = 6\text{ cm}

Subtracting

s

from both sides yields

2s = 12

, so

s = 6\text{ cm}

.

4

Calculate the sector area

A = 3(6) = 18\text{ cm}^2

Substituting

s = 6

into

A = 3s

yields an area of

18\text{ cm}^2

.

Key Concept

Perimeter and Area of a Circular Sector

Question 77Question →

A rhombus has diagonals of lengths $12\text{ cm}$ and $16\text{ cm}$ . What is the perimeter of the rhombus in $\text{cm}$ ?

Show answer & explanation

Answer: 40

Answer

The perimeter of the rhombus is

40\text{ cm}

.

The diagonals of a rhombus bisect each other at right angles, dividing the rhombus into four congruent right-angled triangles. Each triangle has legs measuring

6\text{ cm}

and

8\text{ cm}

. Applying the Pythagorean theorem, the hypotenuse (which is the side length of the rhombus) is

\sqrt{6^2 + 8^2} = \sqrt{100} = 10\text{ cm}

. Since all four sides of a rhombus are equal, the perimeter is

4 \times 10 = 40\text{ cm}

.

Step-by-Step Solution

1

Calculate the lengths of the semi-diagonals

The semi-diagonals are

6\text{ cm}

and

8\text{ cm}

.

The diagonals of a rhombus bisect each other perpendicularly.

2

Determine the side length of the rhombus using the Pythagorean theorem

Side length

s = \sqrt{6^2 + 8^2} = \sqrt{100} = 10\text{ cm}

.

Each side of the rhombus forms the hypotenuse of a right-angled triangle formed by the semi-diagonals.

3

Calculate the total perimeter

Perimeter

P = 4 \times 10 = 40\text{ cm}

.

All four sides of a rhombus are equal in length.

Key Concept

Perimeter of a rhombus derived from diagonal lengths using right-triangle properties

Estimated Time:1m 0s

Question 78Question →

Line $AB$ is parallel to line $CD$ . A transversal line $EF$ intersects line $AB$ at point $P$ and line $CD$ at point $Q$ . If $\angle APQ = (4x - 10)^\circ$ and $\angle PQD = (2x + 30)^\circ$ are alternate interior angles, what is the measure of $\angle BPQ$ ?

110^\circ

70^\circ

20^\circ

100^\circ

Show answer & explanation

Answer: $110^\circ$

Answer

110^\circ

Since alternate interior angles are equal,

4x - 10 = 2x + 30

, yielding

x = 20

. Thus,

\angle APQ = 70^\circ

. Because

\angle APQ

and

\angle BPQ

are adjacent angles on straight line

AB

, their sum is

180^\circ

, giving

\angle BPQ = 110^\circ

.

Step-by-Step Solution

1

Set up the equation for alternate interior angles.

4x - 10 = 2x + 30

Alternate interior angles formed by a transversal cutting parallel lines are equal.

2

Solve the linear equation for

x

.

2x = 40 \implies x = 20

Subtract

2x

and add

10

to both sides.

3

Calculate the measure of

\angle APQ

.

\angle APQ = 4(20) - 10 = 70^\circ$

Substitute

x = 20

into the expression

(4x - 10)^\circ

.

4

Determine

\angle BPQ

using the straight line angle property.

\angle BPQ = 180^\circ - 70^\circ = 110^\circ$

Angles

\angle APQ

and

\angle BPQ

form a linear pair on straight line

AB

, summing to

180^\circ

.

Key Concept

Alternate interior angles of parallel lines and angles on a straight line

Question 79Question →

A rectangular garden measuring $12\text{ m}$ by $8\text{ m}$ has a paved path of uniform width $x\text{ m}$ constructed inside it along its perimeter. If the area of the remaining inner garden is $60\text{ m}^2$ , what is the width $x$ of the path?

1\text{ m}

2\text{ m}

1.5\text{ m}

3\text{ m}

Show answer & explanation

Answer: $1\text{ m}$

Answer

The width of the path is

1\text{ m}

.

The total outer garden area is

12 \times 8 = 96\text{ m}^2

. Subtracting the path width

x

from both ends gives inner dimensions

(12 - 2x)

and

(8 - 2x)

. Equating the inner area

(12 - 2x)(8 - 2x) = 60

yields

x^2 - 10x + 9 = 0

. The realistic physical solution is

x = 1\text{ m}

.

Step-by-Step Solution

1

Express the inner dimensions in terms of path width

x

Length =

(12 - 2x)\text{ m}

, Width =

(8 - 2x)\text{ m}

The path reduces each side dimension by

x

at both ends.

2

Set up the area equation for the inner rectangular garden

(12 - 2x)(8 - 2x) = 60

The area of a rectangle is length multiplied by width.

3

Expand and simplify the quadratic equation

96 - 40x + 4x^2 = 60 \implies 4x^2 - 40x + 36 = 0 \implies x^2 - 10x + 9 = 0

Divide the whole equation by 4 to simplify quadratic terms.

4

Solve for

x

by factoring

(x - 1)(x - 9) = 0 \implies x = 1

or

x = 9

Since the width

x

cannot exceed half of the smaller side (

x < 4\text{ m}

),

x = 9

is extraneous.

Key Concept

Perimeter and Area of Rectangles with Uniform Borders

Estimated Time:1m 30s

Question 80Question →

What is the exact simplified value of $\sin 60^\circ + \cos 45^\circ$ ?

\frac{\sqrt{3} + \sqrt{2}}{2}

\frac{\sqrt{5}}{2}

\frac{\sqrt{3} + 1}{2}

\frac{\sqrt{3} + \sqrt{2}}{4}

Show answer & explanation

Answer: $\frac{\sqrt{3} + \sqrt{2}}{2}$

Answer

\frac{\sqrt{3} + \sqrt{2}}{2}

Substituting the exact values

\sin 60^\circ = \frac{\sqrt{3}}{2}

and

\cos 45^\circ = \frac{\sqrt{2}}{2}

gives

\frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} = \frac{\sqrt{3} + \sqrt{2}}{2}

.

Step-by-Step Solution

1

Identify the exact trigonometric values for the special angles.

\sin 60^\circ = \frac{\sqrt{3}}{2}

and

\cos 45^\circ = \frac{\sqrt{2}}{2}

Standard special angle values in trigonometry.

2

Substitute the values into the given expression.

\frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2}

Direct substitution of known ratio values.

3

Combine the fractions over the common denominator of

2

.

\frac{\sqrt{3} + \sqrt{2}}{2}

Adding numerators over a shared common denominator.

Key Concept

Special Angles and Exact Trigonometric Values

Geometry and Trigonometry

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution