Answer:
The area of LQM is [tex]48cm^2[/tex]
Step-by-step explanation:
Given
Area of PNQ = 8
Area of LPQ = 16
See attachment for triangles
The area of PNQ is calculated as:
[tex]Area = \frac{1}{2} * PQ * PN[/tex]
Substitute 8 for Area
[tex]8 = \frac{1}{2} * PQ * PN[/tex]
[tex]PQ * PN = 16[/tex]
The area of LPQ is calculated as:
[tex]Area = \frac{1}{2} * PQ * LP[/tex]
Substitute 16 for Area
[tex]16= \frac{1}{2} * PQ * LP[/tex]
From the attachment:
[tex]PN + LP =LN[/tex]
Make LP the subject
[tex]LP = LN -PN[/tex]
So:
[tex]16= \frac{1}{2} * PQ * (LN -PN)[/tex]
We have:
[tex]16= \frac{1}{2} * PQ * (LN -PN)[/tex] and [tex]PQ * PN = 16[/tex]
Equate both expressions:
[tex]\frac{1}{2} * PQ *(LN - PN) = PQ * PN[/tex]
Divide both sides by PQ
[tex]\frac{1}{2} (LN - PN) = PN[/tex]
Multiply both sides by 2
[tex]LN - PN = 2PN[/tex]
[tex]LN= 3PN[/tex]
Since PNQ is similar to LNM, the following equivalent ratios exist:
[tex]\frac{LM}{PQ} = \frac{LN}{PN}[/tex]
Substitute [tex]LN= 3PN[/tex]
[tex]\frac{LM}{PQ} = \frac{3PN}{PN}[/tex]
[tex]\frac{LM}{PQ} = 3[/tex]
[tex]LM = 3PQ[/tex]
Area of LQM is:
[tex]Area = \frac{1}{2} * LM * LP[/tex]
This gives:
[tex]Area = \frac{1}{2} * 3PQ * LP[/tex]
[tex]Area = 3 *\frac{1}{2} *PQ * LP[/tex]
Recall that:
[tex]16= \frac{1}{2} * PQ * LP[/tex]
So:
[tex]Area = 3 *16[/tex]
[tex]Area = 48cm^2[/tex]
