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Difficulty: HardTriangles and Pythagorean Theorem

Read the geometric scenario below and calculate the missing perimeter. Enter only the numerical value.

Answer:In △PQR\triangle PQR, an altitude PSPS is drawn to the base QRQR, creating two right triangles, △PQS\triangle PQS and △PSR\triangle PSR. If the length of side PQPQ is 1313 units, the length of side PRPR is 1515 units, and the altitude PSPS is 1212 units, the perimeter of the entire △PQR\triangle PQR is 【42】 units.

Answer

42
To find the perimeter of △PQR\triangle PQR, we need the lengths of its three outer sides: PQPQ, PRPR, and QRQR. We are given PQ=13PQ = 13 and PR=15PR = 15. The altitude PSPS divides QRQR into two segments, QSQS and SRSR, forming two right triangles. Using the Pythagorean theorem in △PQS\triangle PQS, we find QS=132−122=25=5QS = \sqrt{13^2 - 12^2} = \sqrt{25} = 5. Using the Pythagorean theorem in △PSR\triangle PSR, we find SR=152−122=81=9SR = \sqrt{15^2 - 12^2} = \sqrt{81} = 9. The total base is QR=QS+SR=5+9=14QR = QS + SR = 5 + 9 = 14. The perimeter is the sum of the outer sides: 13+15+14=4213 + 15 + 14 = 42.

Step-by-Step Solution

1
Use the Pythagorean theorem in right △PQS\triangle PQS to find the length of base segment QSQS.
QS=132−122=169−144=25=5QS = \sqrt{13^2 - 12^2} = \sqrt{169 - 144} = \sqrt{25} = 5
The altitude PSPS forms a right angle with the base, making PQPQ the hypotenuse. The missing leg is found by subtracting the square of the altitude from the square of the hypotenuse.
2
Use the Pythagorean theorem in right △PSR\triangle PSR to find the length of base segment SRSR.
SR=152−122=225−144=81=9SR = \sqrt{15^2 - 12^2} = \sqrt{225 - 144} = \sqrt{81} = 9
Similarly, PRPR is the hypotenuse of the second right triangle, so the missing leg is found by subtracting the square of the altitude from the square of this hypotenuse.
3
Calculate the total length of the base QRQR.
QR=QS+SR=5+9=14QR = QS + SR = 5 + 9 = 14
The entire base QRQR is the sum of the two segments created by the altitude.
4
Calculate the perimeter of the full △PQR\triangle PQR.
Perimeter = PQ+PR+QR=13+15+14=42PQ + PR + QR = 13 + 15 + 14 = 42
The perimeter is the total length of the outer boundary of the triangle, which is the sum of its three main sides.

Key Concept

Applying the Pythagorean theorem in multi-step geometric figures to find missing side lengths and perimeter.
Estimated Time:2m 0s
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Topics

Triangles and Pythagorean Theorem

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