Solve this---**Extraction Content:**
**Geometric Figure Description:**
* Type: Geometric diagram (Triangle)
* Main Elements:
* A triangle ABC is shown with a right angle at vertex B (∠ABC = 90°).
* Point A is the top vertex, B is the bottom-left vertex, and C is the bottom-right vertex.
* A line segment AQ is drawn from vertex A to point Q on side BC.
* A line segment CP is drawn from vertex C to point P on side AB.
* Segments AQ and CP intersect inside the triangle.
* The angle at vertex A is bisected by segment AQ, indicated by the condition ∠BAQ = ∠QAC and shading.
* The angle at vertex C is bisected by segment CP, indicated by the condition ∠BCP = ∠PCA and shading.
* A square symbol at vertex B denotes a right angle.
* Labels A, B, C, P, and Q identify the points.
**Given Information:**
* ∠BAQ = ∠QAC
* ∠BCP = ∠PCA
* AQ = 9
* CP = 8√2
**Question/Value to Find:**
* AC = ?
**Source/Author:**
* MIND YOUR DECISIONS
* PRESH TALWALKAR
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答案文本
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We have a right triangle ABC with a right angle at B. AQ is the angle bisector of angle A, and CP is the angle bisector of angle C. We're given that AQ equals 9 and CP equals 8 root 2. We need to find the length of AC.
Let's set up a coordinate system to solve this problem systematically. We'll place B at the origin, A on the positive y-axis at coordinates zero comma a, and C on the positive x-axis at coordinates b comma zero. This gives us the length of AC as the square root of a squared plus b squared.
Now we'll apply the Angle Bisector Theorem. When AQ bisects angle BAC, the ratio BQ to QC equals the ratio AB to AC. Similarly, when CP bisects angle BCA, the ratio BP to PA equals the ratio BC to CA. These relationships will help us find the exact coordinates of points P and Q.
Now let's set up our equations. Using the angle bisector theorem, we can find the coordinates of Q and P. Point Q is at ab over a plus c comma zero, and point P is at zero comma ab over b plus c, where c equals the square root of a squared plus b squared. The length AQ equals ac over a plus c, which equals 9. The length CP equals bc over b plus c, which equals 8 root 2.
Now let's solve our system of equations. From ac over a plus c equals 9 and bc over b plus c equals 8 root 2, we can rearrange to get c equals 9a over a minus 9, and c equals 8 root 2 b over b minus 8 root 2. Setting these equal and cross-multiplying gives us a relationship between a and b. After careful algebraic manipulation, we find that AC equals 15.