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Question Video: Finding the Perimeter of a Quadrilateral Using the Triangle Midsegment Theorem Mathematics

Given that 𝐷 and 𝐸 are the midpoints of line segments 𝐴𝐡 and 𝐴𝐢 respectively, 𝐴𝐷 = 32 cm, 𝐴𝐸 = 19 cm, and 𝐷𝐸 = 39 cm, find the perimeter of 𝐷𝐡𝐢𝐸.

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Video Transcript

Given that 𝐷 and 𝐸 are the midpoints of line segments 𝐴𝐡 and 𝐴𝐢, respectively, 𝐴𝐷 equals 32 centimeters, 𝐴𝐸 equals 19 centimeters, and 𝐷𝐸 equals 39 centimeters, find the perimeter of 𝐷𝐡𝐢𝐸.

We can begin by filling in the given length information on the figure of 32 centimeters for 𝐴𝐷, 19 centimeters for 𝐴𝐸, and 39 centimeters for 𝐷𝐸. We can also identify the information that 𝐷 and 𝐸 are midpoints of their respective line segments. So, 𝐸𝐢 also has a length of 19 centimeters, and 𝐷𝐡 has a length of 32 centimeters. We know that we need to work out the perimeter of this quadrilateral 𝐷𝐡𝐢𝐸. We’ve worked out three of these four sides, so we still need to work out the length of the line segment 𝐢𝐡. To do this, let’s use the triangle midsegment theorem.

This theorem states that the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half its length. This means we can recognize that line segment 𝐢𝐡 must be parallel to line segment 𝐷𝐸, and we know something about the length of line segment 𝐢𝐡. Since 𝐷𝐸 is half the length of 𝐢𝐡, we can also write that 𝐢𝐡 equals two times 𝐷𝐸. We were given that 𝐷𝐸 is 39 centimeters, so doubling this, we can calculate that 𝐢𝐡 is 78 centimeters.

So, we now have enough information to calculate the perimeter of 𝐷𝐡𝐢𝐸. Recall that the perimeter is the distance around the outside edge of a shape. So that means we need to add the four lengths of 39, 32, 78, and 19 centimeters, which gives us the final answer that the perimeter of 𝐷𝐡𝐢𝐸 is 168 centimeters.

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