A triangle in the standard (x, y) coordinate plane has vertices at A(1, 1), B(10, 16), and… — ACT

A triangle in the standard $(x, y)$ coordinate plane has vertices at $A(1, 1)$ , $B(10, 16)$ , and $C(5, 9)$ . A line passes through the point $P(4, 7)$ on the side $AC$ and intersects the side $AB$ at a point $Q$ . If this line divides the triangle into two regions of equal area, what is the length of the line segment $PQ$ ?

Answer: 5

Answer

The total area of the triangle is 6. To divide the triangle into two equal-area regions, each region must have an area of 3. Since point

P(4,7)

lies

\frac{3}{4}

of the way from

A

C

, the area of the sub-triangle

\triangle PBC

is only

\frac{1}{4} \times 6 = 1.5

. Thus, the dividing line must intersect side

AB

at a point

Q

to form

\triangle APQ

with an area of 3. Using the area ratio formula,

\text{Area}(\triangle APQ) = \frac{AP}{AC} \times \frac{AQ}{AB} \times \text{Area}(\triangle ABC) \implies 3 = \frac{3}{4} \times \frac{AQ}{AB} \times 6

, which gives

\frac{AQ}{AB} = \frac{2}{3}

. Using the section formula, the coordinates of

Q

are

A + \frac{2}{3}(B - A) = (1, 1) + \frac{2}{3}(9, 15) = (7, 11)

. Finally, the length of

PQ

\sqrt{(7-4)^2 + (11-7)^2} = \sqrt{3^2 + 4^2} = 5

Step-by-Step Solution

Calculate the area of the entire triangle

ABC

using the Shoelace formula.

Area of

\triangle ABC = 6

Establishing the total area of the triangle is necessary to determine the target area of 3 for each of the two equal-area regions.

Determine which side of the triangle the dividing line intersects by comparing the area of

\triangle PBC

to the target area of 3.

The line must intersect side

AB

at a point

Q

Since

P(4,7)

lies

\frac{3}{4}

of the way along

AC

, the base

PC

\frac{1}{4}

AC

. The area of

\triangle PBC

\frac{1}{4} \times 6 = 1.5

. Since this is less than 3, the dividing line cannot intersect side

BC

and must intersect side

AB

instead.

Set up the area ratio equation for

\triangle APQ

to find the ratio

\frac{AQ}{AB}

\frac{AQ}{AB} = \frac{2}{3}

The area of

\triangle APQ

is given by

\text{Area}(\triangle APQ) = \frac{AP}{AC} \times \frac{AQ}{AB} \times \text{Area}(\triangle ABC) \implies 3 = \frac{3}{4} \times \frac{AQ}{AB} \times 6 \implies \frac{AQ}{AB} = \frac{2}{3}

Find the coordinates of

Q

using the section formula along segment

AB

from

A(1,1)

B(10,16)

Q(7, 11)

Applying

Q = A + \frac{2}{3}(B - A) = (1, 1) + \frac{2}{3}(9, 15) = (7, 11)

yields the exact coordinates of

Q

Calculate the length of segment

PQ

using the distance formula between

P(4,7)

and

Q(7,11)

PQ = 5

The question asks for the length of the segment

PQ

, which is the distance between these two points.

Key Concept

Using coordinate geometry formulas and area ratios to solve problems involving geometric figures on the coordinate plane.

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