2017 AMC 8 Problem 22

Area Decomposition Method

We can find the radius by splitting the large triangle into two smaller triangles. The radius will serve as the height for both smaller triangles.

Step 1: Find the length of the hypotenuse

By the Pythagorean theorem, we can calculate the length of the hypotenuse $AB$ of the right triangle $ABC$.

Step 2: Identify the center and radius of the semicircle

Let $O$ be the center of the semicircle. The problem states the semicircle’s straight edge lies on $AC$ and it touches $BC$ at vertex $C$. This means $O$ lies on segment $AC$, and the distance from $O$ to $C$ is exactly the radius $r$.

Step 3: Split the triangle into two smaller areas

Draw a line from the center $O$ to vertex $B$. This line divides the large triangle $ABC$ into two smaller triangles: $△OBC$ and $△OAB$. The semicircle is tangent to $AB$ at a point $T$. Draw the radius $OT$. Because a tangent line is perpendicular to the radius at the point of tangency, $OT$ acts as the height of $△OAB$ to the base $AB$. Because $\angle C=90^{\circ }$, $OC$ acts as the height of $△OBC$ to the base $BC$.

Step 4: Set up the area equation

The total area of $△ABC$ must be equal to the sum of the areas of $△OBC$ and $△OAB$. We calculate each area using the standard triangle area formula and set up an equation.

Step 5: Solve for the radius

Factor out the radius and simplify the equation to find the final value.

Similar Triangles Method

The radius drawn to the point of tangency creates a small right triangle that shares an angle with the large right triangle, making them similar.

Step 1: Set up the side lengths and right angles

Let $O$ be the center of the semicircle on $AC$, and let $r$ be its radius. Since the full length of $AC$ is $12$ and $OC=r$, the remaining segment $OA$ is $12-r$. Draw the radius $OT$ to the point of tangency on the hypotenuse $AB$.

Step 2: Identify the similar triangles

Because $AB$ is tangent to the semicircle at $T$, the angle $\angle ATO$ is $90^{\circ }$. The original large triangle has a right angle at $C$. Since $△ATO$ and $△ACB$ also share the angle at vertex $A$, they have two identical angles. This makes them similar by Angle-Angle similarity.

Step 3: Set up the ratio of corresponding sides

In similar triangles, the ratio of corresponding sides is equal. We will compare the shorter leg to the hypotenuse in both triangles. The hypotenuse of the large triangle is found using the Pythagorean theorem.

Step 4: Solve for the radius

Cross-multiply to clear the denominators, then solve the resulting linear equation for $r$.

Tangent Lengths Method

If we imagine the semicircle as a full circle, point $B$ has two tangent line segments to the circle. These tangent segments must be equal in length.

Step 1: Use the two-tangent theorem

Imagine completing the semicircle into a full circle with center $O$. Because $\angle C=90^{\circ }$, the leg $BC$ is tangent to the circle at $C$. The hypotenuse $AB$ is tangent to the circle at some point $T$. Since tangents from a single external point to a circle are equal in length, the segment $BT$ must equal $BC$.

Step 2: Find the remaining length on the hypotenuse

The full length of the hypotenuse is $AB=13$. Since we just found that $BT=5$, we can subtract this from the total to find the length of the remaining segment $AT$.

Step 3: Apply the Pythagorean theorem to the small triangle

Examine the right-angled $△ATO$. Its legs are $AT$ and the radius $OT$, and its hypotenuse is $OA$. We can write this relationship using the Pythagorean theorem.

Step 4: Solve the algebraic equation

Expand the squared binomial on the right side. The $r^2$ terms appear on both sides and cancel out, leaving a simple linear equation.

Coordinate Geometry Method

By placing the triangle on a coordinate plane, we can use the formula for the perpendicular distance from a point to a line to find the radius.

Step 1: Set up a coordinate system

Place the right angle vertex $C$ at the origin. Since $AC=12$ and $BC=5$, place vertex $A$ on the positive x-axis and vertex $B$ on the positive y-axis. The center $O$ of the semicircle lies on segment $AC$, so its coordinates are $(r,0)$, where $r$ is the unknown radius.

Step 2: Find the equation of the line AB

The line representing the hypotenuse $AB$ passes through $(12,0)$ and $(0,5)$. We can write its equation using the intercept form and then convert it into standard form $Ax+By+C=0$.

Step 3: Apply the point-to-line distance formula

The semicircle is tangent to $AB$, which means the perpendicular distance from its center $O(r,0)$ to the line $5x+12y-60=0$ must exactly equal the radius $r$.

Step 4: Solve the distance equation

Simplify the denominator. Since the center lies on the segment $AC$, the radius $r$ must be less than $12$. Thus, the expression inside the absolute value, $5r-60$, is negative. We take its opposite to remove the absolute value bars.