Question

In ∆PQR, PQ = 39 cm and PN is an altitude. Find PR if QN = 36 cm and RN = 8 cm.

Answer 1

17 cm

Explanation:

We must first find the length of the height, PN. Since PN is an altitude, it makes a right angle with QR; this means that PNQ will be a right triangle, as will PNR. This means we will use the Pythagorean theorem:

a²+b² = c²

Letting h represent PR (since it is the height),

h²+36² = 39²

h²+1296 = 1521

Subtract 1296 from each side:

h²+1296-1296 = 1521-1296

h² = 225

Take the square root of each side:

√(h²) = √(225)

h = 15

PN is 15 cm.

Now we will use it and the other "base," RN, to find PR:

15²+8² = x²

225+64 = x²

289 = x²

Take the square root of each side:

√(289) = √(x²)

17 = x

Answer 2

Use the Pythagorean theorem two times:

`NQ^2+NP^2=QP^2\\\\36^2+h^2=39^2\\\\1296+h^2=1521\qquad\text{subtract 1521 from both sides}\\\\h^2=225\to h=√(225)\\\\\boxed{h=15\ cm}`

second time:

`PR^2=RN^2+NP^2\\\\x^2=8^2+15^2\\\\x^2=64+225\\\\x^2=289\to x=√(289)\\\\\boxed{x=17\ cm}`

Answer: PR = 17 cm.