Question

In a triangle, the altitudes have length 5, 6, and 7. The ratio of the longest side to the shortest side is p/q in lowest terms. What is 2p+q?

a) 3
b) 8
c) 5
d) 10

Answer 1

The value of 2p+q is 8.

Step-by-step explanation:

Let the sides of the triangle be a, b, and c, where a is the shortest side and c is the longest side. According to the properties of altitudes in a triangle, the product of the altitude and the side to which it corresponds is constant. This implies 5a = 6b = 7c.

Using these relations, let's find the ratio of the longest side to the shortest side:

From 5a = 7c, we get a = 7c/5.

From 5a = 6b, we get b = 5a/6 = 7c/6.

So, the ratio of the longest side to the shortest side is c/a = c / (7c/5) = 5.

Therefore, the ratio of the longest side to the shortest side is 5. Hence, 2p + q = 2*5 + 3 = 10 + 3 = 8. Therefore, the value of 2p+q is 8.

In summary, utilizing the properties of altitudes in a triangle and the relation between the altitudes and the sides, we established the ratio between the longest and shortest sides as 5. Consequently, using this ratio to determine 2p + q, the final value is 8.