Question

1. solve for the missing side length.

x =
triangle with vertex b, base ac (with a perpendicular segment from b to ac, creating a right angle, segment length 8, side ab (or bc? wait, the triangle has side from b to a labeled 10, and the perpendicular segment is 8, and ac is split into two equal parts (marked with ticks) with the right segment being x? wait, the diagram: triangle abc, with b connected to a point on ac (lets say d) with bd perpendicular to ac, bd=8, ab=10, and ad=dc (since ac has two ticks, so d is the midpoint). so we need to find x, which is dc (or ad? wait, the x is labeled on the lower part, so maybe dc? wait, the problem is to solve for x, the missing side length. so using pythagoras: in right triangle abd, ad = sqrt(ab² - bd²) = sqrt(10² - 8²) = sqrt(100 - 64) = sqrt(36) = 6. then since d is the midpoint (ticks on ac), ac = 2*ad = 12? wait, no, maybe x is dc? wait, the diagram shows ac with two ticks, so ad = dc, and bd is perpendicular. so ad = sqrt(10² - 8²) = 6, so dc = 6? wait, but maybe the triangle is isoceles? wait, the original triangle: ab = bc? wait, the side from b to a is 10, and from b to c? wait, no, the diagram: vertex b, base ac, with bd perpendicular to ac, bd=8, ab=10, and ac has two equal segments (ad and dc, marked with ticks), so ad = dc = x? wait, no, the x is labeled on the lower part, so maybe dc is x? wait, the problem is to find x. so in right triangle abd, ad = sqrt(ab² - bd²) = sqrt(10² - 8²) = 6. then since ad = dc (because ac is bisected by bd, as indicated by the ticks), so dc = 6, so x = 6? wait, but maybe the triangle is isoceles with ab = bc? wait, no, ab is 10, bc would be 10, but bd is 8, so dc would be sqrt(bc² - bd²) = sqrt(10² - 8²) = 6. so x is 6. so the ocr text is: 2. solve for the missing side length.
x =
diagram: triangle abc, b to ac is perpendicular (bd), bd=8, ab=10, ac has two ticks (ad=dc), x is dc (or ad? the x is labeled on the lower part, so dc is x? wait, the diagram shows x as the length from d to c, with d being the foot of the perpendicular from b to ac. so ad and dc are equal (ticks), so ad = dc = x? wait, no, ad is sqrt(10² - 8²) = 6, so dc is also 6, so x = 6. so the ocr text is the problem statement and diagram: 2. solve for the missing side length.
x =
triangle with b, a, c; bd perpendicular to ac, bd=8, ab=10, ac has two equal segments (ad and dc), x is dc (or ad? the x is marked on the lower segment, so dc is x).

Explanation:

Step1: Identify the triangle type

The triangle is isosceles (since the segments on side \( AC \) are equal), and the line from \( B \) to \( AC \) is a perpendicular bisector, forming two right triangles.

Step2: Apply the Pythagorean theorem

In the right triangle with hypotenuse \( 10 \) and one leg \( 8 \), let the other leg (half of \( AC \)) be \( y \). Then \( y^2 + 8^2 = 10^2 \).
\[
y^2 + 64 = 100
\]
\[
y^2 = 100 - 64 = 36
\]
\[
y = 6
\]

Step3: Find \( x \)

Since \( x \) is equal to \( y \) (because the perpendicular bisects \( AC \) into two equal parts), \( x = 6 \).

Answer:

\( 6 \)