In triangle ABC, points P and Q are on side \(\overline{AB},\) and point R is on side \(\overline{AC}.\) If \(AP = \frac{PQ}{3} = \frac{QB}{2}\) and \(\frac{AR}{2} = \frac{RC}{3},\) then find \(\frac{[QBC]}{[CRQ]}.\)

I'm getting \(\frac{6}{5}\) but apparently it's **incorrect**. Please, anyone, I really need help. Thank you!

readyformathlearning Jul 17, 2024

#1**+2 **

Analyzing the Problem

We are given a triangle ABC with points P, Q, and R on sides AB, AB, and AC respectively. Specific ratios are provided for segment lengths related to these points. We need to find the ratio of the areas of triangles QBC and CRQ.

Solution Strategy

Relate Segment Lengths: Use the given ratios to express the lengths of AP, PQ, QB, AR, and RC in terms of a single variable.

Express Triangle Areas: Formulate the areas of triangles QBC and CRQ using the segment lengths obtained in step 1.

Find the Ratio of Areas: Divide the area of triangle QBC by the area of triangle CRQ and simplify the expression.

Steps to Solve

Relating Segment Lengths:

We are given: AP = PQ / 3 = QB / 2 and AR / 2 = RC / 3

Let x be the length of AP (or PQ / 3 or QB / 2).

Then, PQ = 3x and QB = 2x.

Similarly, let y be the length of AR (or 2 * RC / 3).

Then, RC = 3y / 2.

Expressing Triangle Areas:

Area of triangle QBC = (1/2) * Base * Height = (1/2) * QB * RC = (1/2) * 2x * (3y / 2) = 3xy

Area of triangle CRQ = (1/2) * Base * Height = (1/2) * CR * PQ = (1/2) * (3y / 2) * 3x = 4.5xy

Ratio of Areas:

[QBC] / [CRQ] = (3xy) / (4.5xy) = 2 / 3

Answer

Therefore, the ratio of the areas of triangles QBC and CRQ is 2:3.

bader Jul 17, 2024