Charlene puts together two isosceles triangles so that they share a base, creating a kite. The legs of the triangles are 10 inches and 17 inches, respectively. If the length of the base for both triangles is 16 inches long, what is the length of the kite's other diagonal?
Answer (2)
Analyzes the kite as two isosceles triangles sharing a base.
Calculates the altitude h 1 of the first triangle using the Pythagorean theorem: h 1 = 1 0 2 − 8 2 = 6 inches.
Calculates the altitude h 2 of the second triangle using the Pythagorean theorem: h 2 = 1 7 2 − 8 2 = 15 inches.
Sums the altitudes to find the length of the kite's other diagonal: h 1 + h 2 = 6 + 15 = 21 inches.
Explanation
Problem Analysis Let's analyze the problem. We have a kite formed by two isosceles triangles sharing a common base. The legs of the triangles are 10 inches and 17 inches, and the common base is 16 inches. We need to find the length of the kite's other diagonal, which is the sum of the altitudes of the two triangles to the common base.
Altitude of First Triangle Let's calculate the altitude of the first isosceles triangle (with legs of 10 inches). Let the altitude be h 1 . We can use the Pythagorean theorem to find h 1 . The base is 16 inches, so half of the base is 8 inches. Thus, h 1 = 1 0 2 − 8 2 = 100 − 64 = 36 = 6 So, h 1 = 6 inches.
Altitude of Second Triangle Now, let's calculate the altitude of the second isosceles triangle (with legs of 17 inches). Let the altitude be h 2 . Again, we use the Pythagorean theorem. Half of the base is 8 inches. Thus, h 2 = 1 7 2 − 8 2 = 289 − 64 = 225 = 15 So, h 2 = 15 inches.
Length of the Other Diagonal The length of the kite's other diagonal is the sum of the altitudes of the two triangles: h 1 + h 2 = 6 + 15 = 21 Therefore, the length of the kite's other diagonal is 21 inches.
Examples
Kites are a classic example of geometry in action! Understanding the properties of shapes like triangles and kites helps us design and build them. Knowing how to calculate diagonals and altitudes allows us to optimize kite designs for better flight and stability. This same knowledge can be applied to architecture, engineering, and even art, where understanding shapes and their properties is crucial for creating functional and aesthetically pleasing structures.
The length of the kite's other diagonal, formed by two isosceles triangles, is calculated by finding the altitudes of each triangle. The first triangle's altitude is 6 inches, while the second triangle's altitude is 15 inches. Adding these together gives a total length of 21 inches for the kite's other diagonal.
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