In the xy-plane, right triangle ABC has vertices at A(0, 0), B(k, 0), and C(0, 6), where k… — SAT

In the $xy$ -plane, right triangle $ABC$ has vertices at $A(0, 0)$ , $B(k, 0)$ , and $C(0, 6)$ , where $k$ is a positive constant. The right angle of the triangle is at vertex $A$ . If the length of the altitude from $A$ to side $BC$ is $4.8$ , what is the value of $k$ ?

A
3.6
B
4.5
8Answer
D
10

Answer

The correct value is 8. The area of right triangle

ABC

is given by

\frac{1}{2} \times \text{base} \times \text{height}

. Using the legs, the area is

\frac{1}{2} \times k \times 6 = 3k

. Using the hypotenuse and the altitude, the area is

\frac{1}{2} \times BC \times 4.8 = 2.4 \times BC

. Equating the two expressions gives

3k = 2.4 \times BC

, which simplifies to

BC = 1.25k

. Applying the Pythagorean theorem,

(1.25k)^2 = k^2 + 6^2 \implies 1.5625k^2 = k^2 + 36 \implies 0.5625k^2 = 36 \implies k^2 = 64

. Since

k

must be positive,

k = 8

Step-by-Step Solution

Express the area of right triangle

ABC

in two different ways using its base and height.

Area =

3k

and Area =

2.4 \times BC

The area of a right triangle can be calculated using the legs as the base and height, Area =

\frac{1}{2} \times AB \times AC = \frac{1}{2} \times k \times 6 = 3k

. Alternatively, using the hypotenuse

BC

as the base and the altitude to the hypotenuse as the height, Area =

\frac{1}{2} \times BC \times 4.8 = 2.4 \times BC

Equate the two area expressions to find the length of the hypotenuse

BC

in terms of

k

3k = 2.4 \times BC \implies BC = 1.25k

Since both expressions represent the area of the same triangle, they must be equal.

Apply the Pythagorean theorem to right triangle

ABC

to set up an equation for

k

(1.25k)^2 = k^2 + 6^2 \implies 1.5625k^2 = k^2 + 36

In right triangle

ABC

, the Pythagorean theorem states that

BC^2 = AB^2 + AC^2

Solve the equation for

k

0.5625k^2 = 36 \implies k^2 = 64 \implies k = 8

(since

k > 0

Subtracting

k^2

from both sides gives

0.5625k^2 = 36

. Dividing both sides by

0.5625

yields

k^2 = 64

. Taking the square root gives

k = 8

, since

k

represents a positive coordinate.

Key Concept

Relating the area of a right triangle to its legs and hypotenuse altitude to set up proportions, followed by applying the Pythagorean theorem.

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic