solve---**Question Stem:**
5 Find the exact values of a and x.
**Chart/Diagram Description:**
* **Type:** Geometric figure (A large triangle divided into two smaller right-angled triangles by an altitude).
* **Main Elements:**
* The diagram shows a large triangle with an altitude drawn from its top vertex perpendicularly to its base, effectively dividing it into two smaller right-angled triangles.
* **Left Right-Angled Triangle:**
* The hypotenuse is labeled `x cm`.
* The angle at the bottom-left corner of this triangle is `60°`.
* It has a right angle where the altitude meets the base.
* **Right Right-Angled Triangle:**
* The hypotenuse is labeled `a cm`.
* The angle at the bottom-right corner of this triangle is `30°`.
* It also has a right angle where the altitude meets the base.
* **Base Length:** The entire base of the large triangle (the sum of the bases of the two smaller right-angled triangles) is indicated by an arrow and labeled `20 cm`.
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We have a triangle divided by an altitude into two right triangles. The left triangle has a 60-degree angle and hypotenuse x centimeters. The right triangle has a 30-degree angle and hypotenuse a centimeters. The total base length is 20 centimeters. We need to find the exact values of a and x using trigonometric relationships.
The altitude h is the key connecting element between both right triangles. This height is shared by both triangles and serves as the vertical side for each. The left triangle and right triangle both use this same altitude h. Finding this height will be crucial for determining both unknown values a and x.
Let's focus on the left triangle with the 60-degree angle. Using trigonometry, we can establish key relationships. The tangent of 60 degrees equals h over the base, so the base equals h divided by the square root of 3. The sine of 60 degrees equals h over x, so x equals h divided by sine 60 degrees, which simplifies to 2h over the square root of 3.
Now let's analyze the right triangle with the 30-degree angle. Using trigonometry again, the tangent of 30 degrees equals h over the base, so the base equals h times the square root of 3. The sine of 30 degrees equals h over a, so a equals h divided by sine 30 degrees, which simplifies to 2h.
Now we use the constraint that the total base equals 20 centimeters. The sum of both base segments gives us h over square root 3 plus h times square root 3 equals 20. Factoring out h, we get h times the quantity 1 over square root 3 plus square root 3, which simplifies to h times 4 over square root 3 equals 20. Solving for h gives us h equals 5 square root 3 centimeters. Therefore, a equals 2h which is 10 square root 3 centimeters, and x equals 2h over square root 3 which is 10 centimeters.