Answer:
perimeter = 16 ft
Explanation:
You want the perimeter of an isosceles triangle with side length 5 ft and altitude 4 ft.
Base length
The side lengths being equal mean the triangle is isosceles. That also means altitude PS bisects the peak angle and segment QR. Once we find the length QS using the Pythagorean theorem, we can double that measure to find QR.
The Pythagorean theorem tells us ...
QS² +PS² = QP²
QS² +(4 ft)² = (5 ft)²
QS² = (25 ft²) -(16 ft²) = 9 ft² . . . . . evaluate squares, subtract 16 ft²
QS = √(9 ft²) = 3 ft
Now we know the base length is ...
QR = 2·QS = 2(3 ft) = 6 ft
Perimeter
The perimeter is the sum of the outside edge lengths:
Perimeter = PQ +QR +RP
Perimeter = (5 ft) +(6 ft) +(5 ft) = 16 ft
The perimeter of ∆PQR is 16 feet.
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Additional comment
With some practice, you can learn to recognize a 3-4-5 right triangle. That is the triangle used here for each of the right triangles that make up ∆PQR. Once you identify the long side as 4 and the hypotenuse as 5, you know immediately the short side is 3.
The set {3, 4, 5} is called a "Pythagorean triple." Other triples commonly seen in high school problems are {5, 12, 13}, {7, 24, 25}, {8, 15, 17}. Legs of these lengths form a right triangle. Knowing any two (or their ratio) tells you immediately the value of the third. Of course, these triples can be scaled by any factor. (3, 4, 5) scaled by 2 is (6, 8, 10), another commonly seen triple.
There are no isosceles right triangles that have integer values for both the legs and the hypotenuse. If the legs are both 4, the hypotenuse will be 4√2 ≈ 5.657... (an irrational number), not 5.