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# Exploring the properties of Isosceles triangles

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Hello,

Given an isosceles triangle ABC where AB=AC and the base BC is of length 8 cm. The height from A to the base BC divides it into two segments, BD and DC. If the length of BD is 3 cm, what is the length of segment DC and the height of the triangle?

Apr 11, 2024

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We can solve this problem using the properties of isosceles triangles and the Pythagorean theorem.

Isosceles Triangle Properties:

Since AB = AC in an isosceles triangle ABC, angles B and C are congruent.

Dividing the Base:

The height from A to BC is the perpendicular bisector of the base (divides it into two segments of equal length) in an isosceles triangle.

Segment Lengths:

Given that BC = 8 cm and BD = 3 cm, we know DC = BC - BD = 8 cm - 3 cm = 5 cm.

Finding the Height:

We can use the Pythagorean theorem on right triangle ABD (since the height is perpendicular to the base).

Pythagorean Theorem:

a^2 + b^2 = c^2 (where a and b are the lengths of the legs and c is the length of the hypotenuse)

In this case:

a (length of leg BD) = 3 cm

c (length of hypotenuse AB) = unknown (represents the height of the triangle)

b (length of leg AD) = unknown

Solving for Height (c):

b^2 = c^2 - a^2 (rearranging the equation)

b^2 = c^2 - 3^2 (substitute known values)

Since AD is half of the base (property of isosceles triangle):

AD = 1/2 * BC = 1/2 * 8 cm = 4 cm

Substitute the value of AD (b) in the equation:

4^2 = c^2 - 3^2

Simplify:

16 = c^2 - 9

25 = c^2

Take the square root of both sides (remember to consider both positive and negative since squaring either results in 25):

c = ± 5

Positive Height:

Since the height represents a distance, we take the positive value:

c (height of the triangle) = 5 cm

Therefore:

Length of segment DC = 5 cm

Height of the triangle = 5 cm

Apr 14, 2024