Questions: On the star map, three stars form a right triangle with star Vega at the right angle. The distance between Vega and Deneb is 6 centimeters. The distance between Deneb and Altair is 11 centimeters. What is the distance between Vega and Altair on the star map? Round your answer to the nearest tenth. The distance between Vega and Altair on the star map is about centimeters.

Solution Steps

The problem states that the stars form a right triangle with Vega at the right angle. This means the sides connecting Vega to Deneb and Vega to Altair are the legs of the right triangle, and the side connecting Deneb to Altair is the hypotenuse. We are given that the distance between Vega and Deneb is 6 cm, and the distance between Deneb and Altair is 11 cm. We are asked to find the distance between Vega and Altair.

Let \(a\) be the distance between Vega and Deneb, \(b\) be the distance between Vega and Altair, and \(c\) be the distance between Deneb and Altair. According to the Pythagorean theorem, in a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse: \(a^2 + b^2 = c^2\). We have \(a = 6\) cm and \(c = 11\) cm. We want to find \(b\).

Substitute the given values into the Pythagorean theorem: \(6^2 + b^2 = 11^2\) \(36 + b^2 = 121\) \(b^2 = 121 - 36\) \(b^2 = 85\) \(b = \sqrt{85}\) \(b \approx 9.2195\)

We are asked to round the answer to the nearest tenth. Since the hundredths digit is 1, we round down: \(b \approx 9.2\) cm

Final Answer