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In a triangle PQR, the lengths of two sides are given as QR = 12 and
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29 Jan 2025, 13:05
29 Jan 2025, 13:05
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In a triangle PQR, the lengths of two sides are given as QR = 12 and PR = 13. The perimeter of the triangle is 32.
Quantity A |
Quantity B |
Measure of angle Q |
90° |
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Re: In a triangle PQR, the lengths of two sides are given as QR = 12 and
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03 Mar 2025, 10:45
03 Mar 2025, 10:45
Expert Reply
We have assumed the measure of angle Q as x degrees.
As the perimeter of triangle PQR is given as 32 , we get $\(\mathrm{PQ}=32-(13+12)=32-25=7\)$
Now, we have $\(\mathrm{PR}^2=13^2=169\)$ which is less than $\(\mathrm{QR}^2+\mathrm{PQ}^2=12^2+7^2=144+49=193\)$ which implies the measure of angle Q i.e. x is less than 90 degrees.
[Note: - In a triangle say $\(\mathrm{ABC}, \triangle \mathrm{C}^2=\mathrm{AB}^2+\mathrm{BC}^2 \Leftrightarrow \angle \mathrm{~B}=90^{\circ}, \mathrm{AC}^2>\mathrm{AB}^2+\mathrm{BC}^2 \Leftrightarrow \angle \mathrm{~B}>90^{\circ}\)$ $\(\left.\& \mathrm{AC}^2<\mathrm{AB}^2+\mathrm{BC}^2 \Leftrightarrow \angle \mathrm{~B}<90^{\circ}\right]\)$
Hence column $\(B\)$ has higher quantity when compared with column $\(A\)$, so the answer is (B).
As the perimeter of triangle PQR is given as 32 , we get $\(\mathrm{PQ}=32-(13+12)=32-25=7\)$
Now, we have $\(\mathrm{PR}^2=13^2=169\)$ which is less than $\(\mathrm{QR}^2+\mathrm{PQ}^2=12^2+7^2=144+49=193\)$ which implies the measure of angle Q i.e. x is less than 90 degrees.
[Note: - In a triangle say $\(\mathrm{ABC}, \triangle \mathrm{C}^2=\mathrm{AB}^2+\mathrm{BC}^2 \Leftrightarrow \angle \mathrm{~B}=90^{\circ}, \mathrm{AC}^2>\mathrm{AB}^2+\mathrm{BC}^2 \Leftrightarrow \angle \mathrm{~B}>90^{\circ}\)$ $\(\left.\& \mathrm{AC}^2<\mathrm{AB}^2+\mathrm{BC}^2 \Leftrightarrow \angle \mathrm{~B}<90^{\circ}\right]\)$
Hence column $\(B\)$ has higher quantity when compared with column $\(A\)$, so the answer is (B).
gmatclubot
Re: In a triangle PQR, the lengths of two sides are given as QR = 12 and [#permalink]
03 Mar 2025, 10:45
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