91Ó°ÊÓ

Solve each equation by factoring or by using the quadratic formula. The quadratic formula is:

If $a{x}^2+bx+c=0$, with $a≠0$, then$x=\frac{−bÂ\pm \sqrt{b^2−4ac}}{2a}$.

$z^2=4(2z−3)$

For Exercises 23-27 write proofs in paragraph form. (Hint: you can use theorems from this section to write fairly short proofs for exercise 23 and 24.)

Given: $\mathbf{m}∠\mathbf{RTS}\mathbf{=}\mathbf{90}$;$\stackrel{↔}{\mathrm{MN}}$ is the$âŠ\yen$ bisector of $\stackrel{¯}{\mathrm{TS}}$.

Prove:$\stackrel{¯}{\mathrm{TM}}$is a median.

Copy each three-dimensional figure and with coloured pencils outline the triangles listed. What postulate proves that these triangles are congruent?

Given: Cube whose faces are congruent squares.

Show: $\mathbf{â–³}\mathbf{ABF}$,$\mathbf{â–³}\mathbf{BCG}$

Draw an isosceles $\mathrm{Δ}\mathbf{ABC}$whose vertex angle, $∠\mathbf{A}$has measure 80.

a. Draw $\stackrel{¯}{\mathrm{AX}}$the bisector of an exterior angle at $A$Is $\stackrel{¯}{\mathrm{AX}}âˆ\yen \stackrel{¯}{\mathrm{BC}}$? Explain.

b. Would your answer change if the measure of$∠\mathbf{A}$ changed?

Solve each equation by factoring or by using the quadratic formula. The quadratic formula is:

If $a{x}^2+bx+c=0$, with$a≠0$, then$x=\frac{−bÂ\pm \sqrt{b^2−4ac}}{2a}$.

$x(x+5)=14$

Copy each three-dimensional figure and with coloured pencils outline the triangles listed. What postulate proves that these triangles are congruent?

Given: pyramid with square base;$\mathbf{VA}\mathbf{=}\mathbf{VB}\mathbf{=}\mathbf{VC}\mathbf{=}\mathbf{VD}$

Show: $\mathbf{â–³}\mathbf{VAB}$,$\mathbf{â–³}\mathbf{VBC}$

Find the values of$\mathit{x}$ and y.

27. ln equiangular $\mathit{Δ}\mathbf{ABC}$,$\mathbf{AB}\mathbf{=}\mathbf{4}\mathbf{x}\mathbf{-}\mathbf{y}\mathbf{,}\mathbf{BC}\mathbf{=}\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{3}\mathbf{y}$ and $\mathbf{AC}\mathbf{=}\mathbf{7}$.

For Exercises 23-27 write proofs in paragraph form. (Hint: You can use theorems from this section to write fairly short proofs for Exercises 23 and 24.)

Given:$\stackrel{\mathit{¯}}{\mathit{E}\mathit{H}}$and$\stackrel{\mathit{¯}}{\mathit{F}\mathit{J}}$are medians of scalene width="74" height="20" role="math">ΔEFG;P is on$\stackrel{\mathit{→}}{\mathit{E}\mathit{H}}$ such that$\stackrel{\mathbf{¯}}{\mathbf{E}\mathbf{H}}\mathbf{â‰\dots }\stackrel{\mathbf{¯}}{\mathbf{H}\mathbf{P}}\mathbf{;}\mathit{Q}$ is onwidth="22" height="24" role="math">FJ→ such that $\stackrel{\mathit{¯}}{\mathit{F}\mathit{J}}\mathbf{â‰\dots }\stackrel{\mathit{¯}}{\mathit{J}\mathit{Q}}$.

Prove: a. $\stackrel{\mathit{¯}}{\mathit{G}\mathit{Q}}\mathbf{â‰\dots }\stackrel{\mathit{¯}}{\mathit{G}\mathit{P}}$

b. $\stackrel{\mathit{¯}}{\mathit{G}\mathit{Q}}$and $\stackrel{\mathit{¯}}{\mathit{G}\mathit{P}}$are both parallel to$\stackrel{\mathit{¯}}{\mathit{E}\mathit{F}}$

c.$\mathit{P}\mathbf{,}\mathit{G}\mathbf{,}$ and$\mathit{Q}$are collinear.

In Exercises 28-33 xrepresents the length of a segment. When a value of xdoesn't make sense as a length, eliminate that value of x.

$x\left(x−50\right)=0$

Find the values of x and y.

In equilateral$\mathrm{Δ}\mathit{D}\mathit{E}\mathit{F}\mathbf{,}$$\mathit{m}∠\mathit{D}\mathbf{=}\mathit{x}\mathbf{+}\mathit{y}$ and $\mathit{m}∠\mathit{E}\mathbf{=}\mathbf{2}\mathit{x}\mathbf{-}\mathit{y}$.