Answer:
Blank 1: 20
Blank 2: 43.6
Step-by-step explanation:
Question 1
To find the length of leg PQ in right triangle PQR, we can use the Pythagorean Theorem:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Pythagorean Theorem}}\\\\a^2+b^2=c^2\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$a$ and $b$ are the legs of the right triangle.}\\\phantom{ww}\bullet\;\textsf{$c$ is the hypotenuse (longest side) of the right triangle.}\\\end{array}}[/tex]
In this case:
- a = PQ
- b = QR = 21 cm
- c = PR = 29 cm
Substitute the values into the formula and solve for PQ:
[tex]PQ^2+QR^2=PR^2\\\\PQ^2+21^2=29^2\\\\PQ^2+441=841\\\\PQ^2=841-441\\\\PQ^2=400\\\\PQ=\sqrt{400}\\\\PQ=20[/tex]
Therefore, the measure of side PQ is:
[tex]\LARGE\boxed{\boxed{PQ=20\; \sf cm}}[/tex]
[tex]\dotfill[/tex]
Question 2
To find the measure of angle R, we can use the cosine trigonometric ratio:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Cosine trigonometric ratio}}\\\\\sf \cos(\theta)=\dfrac{A}{H}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$\theta$ is the angle.}\\\phantom{ww}\bullet\;\textsf{$A$ is the side adjacent the angle.}\\\phantom{ww}\bullet\;\textsf{$H$ is the hypotenuse (the side opposite the right angle).}\end{array}}[/tex]
In this case:
- θ = R
- A = QR = 21
- H = PR = 29
Substitute the values into the formula and solve for R:
[tex]\cos R=\dfrac{21}{29}\\\\\\R=\cos^{-1}\left(\dfrac{21}{29}\right)\\\\\\R=43.6028189727...\\\\\\R=43.6^{\circ}\; \sf (nearest\;tenth)[/tex]
Therefore, the measure of angle R rounded to the nearest tenth is:
[tex]\LARGE\boxed{\boxed{R=43.6^{\circ}}}[/tex]
