Question Video: Applying Properties of Congruence to Solve Problems | Nagwa Question Video: Applying Properties of Congruence to Solve Problems | Nagwa

Question Video: Applying Properties of Congruence to Solve Problems Mathematics • First Year of Preparatory School

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The two triangles in the given figure are congruent. Work out the area of triangle ๐ด๐ต๐ถ.

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Video Transcript

The two triangles in the given figure are congruent. Work out the area of triangle ๐ด๐ต๐ถ.

Weโ€™re told here that the two triangles are congruent. That means that pairs of corresponding angles will be equal and pairs of corresponding sides will be equal. Weโ€™ll need to use this fact to help us work out the area of triangle ๐ด๐ต๐ถ.

We can recall that to find the area of a triangle, we multiply half times the base times the perpendicular height. When we look at triangle ๐ด๐ต๐ถ, we can see that we donโ€™t know the base length of this triangle, which is why weโ€™ll need to use the fact that this is congruent with triangle ๐ท๐ธ๐น to help us work out the length of ๐ต๐ถ.

In this question, we werenโ€™t given a congruency relationship, so weโ€™ll need to establish which sides correspond to which sides. Letโ€™s start with the hypotenuse, the longest side on triangle ๐ด๐ต๐ถ. This will correspond with the longest side or hypotenuse on our other triangle. So, ๐ด๐ถ and ๐ท๐น will be congruent.

On triangle ๐ด๐ต๐ถ, if we go from the hypotenuse down to the right angle along the line ๐ด๐ต, this corresponds to the same journey or path from the hypotenuse down to the right angle on triangle ๐ท๐ธ๐น. So, ๐ด๐ต and ๐ท๐ธ will be the same length of 5.1. The final pair of sides ๐ต๐ถ and ๐ธ๐น will also be congruent, and theyโ€™ll be of length 4.1.

We now have enough information to work out the area of triangle ๐ด๐ต๐ถ. Filling in the values for our base length of 4.1 and the perpendicular height of 5.1, weโ€™ll have a half times 4.1 times 5.1. We can work out 4.1 times 5.1 by calculating 41 times 51. As our values had a total of two decimal digits, then our answer will also have two decimal digits. Half of 20.91 will give us 10.455. The units here would be square units. This is our answer for the area of triangle ๐ด๐ต๐ถ. Note that if we worked out the area of triangle ๐ท๐ธ๐น instead, we wouldโ€™ve got the same answer as both of these triangles are congruent.

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