Question

LMN is a right-angle triangle. Angle NLM=90. PQ is parallel to LM. The area of triangle PNQ is 8cm^2. The area of triangle LPQ is 16cm^2. Work out the area of triangle

Answer 1

Answer:

The area of LQM is $48cm^2$

Step-by-step explanation:

Given

Area of PNQ = 8

Area of LPQ = 16

See attachment for triangles

The area of PNQ is calculated as:

$Area = \frac{1}{2} * PQ * PN$

Substitute 8 for Area

$8 = \frac{1}{2} * PQ * PN$

$PQ * PN = 16$

The area of LPQ is calculated as:

$Area = \frac{1}{2} * PQ * LP$

Substitute 16 for Area

$16= \frac{1}{2} * PQ * LP$

From the attachment:

$PN + LP =LN$

Make LP the subject

$LP = LN -PN$

So:

$16= \frac{1}{2} * PQ * (LN -PN)$

We have:

$16= \frac{1}{2} * PQ * (LN -PN)$ and $PQ * PN = 16$

Equate both expressions:

$\frac{1}{2} * PQ *(LN - PN) = PQ * PN$

Divide both sides by PQ

$\frac{1}{2} (LN - PN) = PN$

Multiply both sides by 2

$LN - PN = 2PN$

$LN= 3PN$

Since PNQ is similar to LNM, the following equivalent ratios exist:

$\frac{LM}{PQ} = \frac{LN}{PN}$

Substitute $LN= 3PN$

$\frac{LM}{PQ} = \frac{3PN}{PN}$

$\frac{LM}{PQ} = 3$

$LM = 3PQ$

Area of LQM is:

$Area = \frac{1}{2} * LM * LP$

This gives:

$Area = \frac{1}{2} * 3PQ * LP$

$Area = 3 *\frac{1}{2} *PQ * LP$

Recall that:

$16= \frac{1}{2} * PQ * LP$

So:

$Area = 3 *16$

$Area = 48cm^2$