+4 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?
+1133 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
Add 9 to both sides:
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
