Digital SAT Math Practice Question #1198
Test your problem-solving accuracy on this official-caliber item. Select an answer choice to check your reasoning instantly.
How does the radius of a circle relate to the tangent line at the point where the radius meets the circle? What mathematical property can you use to find the slope of the tangent line?
Once you have the equation of the tangent line, how do you find its x-intercept and y-intercept? What kind of triangle is formed by the origin and these two intercepts, and how do you calculate its area?
⚡ Hack the Test: If you have access to a graphing calculator like Desmos, you can quickly visualize and verify the solution. 1. **Graph the circle:** Input \((x-1)^2 + (y-2)^2 = 25\). 2. **Plot the point P:** Plot \((4,6)\). 3. **Find the slope of the radius:** Calculate the slope from \(C(1,2)\) to \(P(4,6)\), which is \((6-2)/(4-1) = 4/3\). 4. **Determine the tangent slope:** The tangent line's slope is the negative reciprocal, \(-3/4\). 5. **Graph the tangent line:** Use the point-slope form \(y - 6 = -3/4(x - 4)\) and graph it. Alternatively, you can use the equation \(3x + 4y = 36\) directly if you've already derived it. 6. **Identify intercepts:** Visually inspect where the tangent line crosses the x-axis (Q) and y-axis (R). The calculator will show \((12,0)\) and \((0,9)\). 7. **Calculate area:** For a right triangle with vertices at the origin and on the axes, the area is \(1/2 \times |x_{intercept}| \times |y_{intercept}| = 1/2 \times 12 \times 9 = 54\). This method allows for quick verification of your algebraic steps or can be used to solve the problem directly if you are proficient with the graphing tool.
To solve this problem, we need to find the equation of the tangent line to the circle at point P, then find its x and y intercepts, and finally calculate the area of the triangle formed by these intercepts and the origin. **Step 1: Identify...
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