Question

In triangle ABC, AD is perpendicular to BC, BE is perpendicular to AC, BC = 24 cm, AD = 18 cm,
AC = 27 cm. Find BE.​

Answer 1

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Because 2 of the sides are perpendicular, the triangle must be a right triangle with AC as hypotenuse. Assuming that point D lies on the hypotenuse, then BD is a segment connecting vertex of the right angle to the hyp. ( at a rt. angle) and is therefore the interior altitude of triangle.

This altitude divides the rt. triangle into 2 smaller rt. triangles which are each similar (geometrically) to the larger rt. triangle. This is proven by seeing that seg. BD divides rt. angle of large triangle into 2 complementary angles. If the 2 acute angles of the larger triangle are: a and b, which must be complementary in a rt. triangle, the 2 acute angle formed by seg. BD are also =a, and b.

The 2 acute, complementary angles in each of the smaller rt. triangles must also be a and b, since each smaller triangle is part of the larger one, and one acute angle of each coincides with an angle of the larger triangle.

In similar figures, the lengths of corresponding sides are in proportion, so that triangle sides AD is to DB as DB is to DC , or 4:x as x: 9 ( 4/x =x/9), and therefore x (length of DB) = 6.

Answer 2

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Because 2 of the sides are perpendicular, the triangle must be a right triangle with AC as hypotenuse. Assuming that point D lies on the hypotenuse, then BD is a segment connecting vertex of the right angle to the hyp. ( at a rt. angle) and is therefore the interior altitude of triangle.

This altitude divides the rt. triangle into 2 smaller rt. triangles which are each similar (geometrically) to the larger rt. triangle. This is proven by seeing that seg. BD divides rt. angle of large triangle into 2 complementary angles. If the 2 acute angles of the larger triangle are: a and b, which must be complementary in a rt. triangle, the 2 acute angle formed by seg. BD are also =a, and b.

The 2 acute, complementary angles in each of the smaller rt. triangles must also be a and b, since each smaller triangle is part of the larger one, and one acute angle of each coincides with an angle of the larger triangle.

In similar figures, the lengths of corresponding sides are in proportion, so that triangle sides AD is to DB as DB is to DC , or 4:x as x: 9 ( 4/x =x/9), and therefore x (length of DB) = 6.