Given: △EDN∼△LKI, DQ, KO are altitudes, DN=12, KI=4, DQ=KO+6. Find: QN and OI.

QN = 3√7 OI = √7

Step-by-step explanation:

The ratio of corresponding sides DN and KI is 12 : 4 = 3 : 1. The same ratio applies to altitudes DQ and KO. Since the difference between these altitudes is 6 and the difference between their ratio units is 3-1 = 2, each ratio unit must stand for 6/2 = 3 units of linear measure. That is, ...

DQ = (3 units) ·3 = 9 units

KO = (3 units) ·1 = 3 units

Then the base lengths QN and OI can be found from the Pythagorean theorem:

KI² = KO² + OI²

4² = 3² + OI²

OI = √ (16 - 9)

OI = √7

QN = 3·OI = 3√7