Apply Algebra And Geometry - Air Force Officer Qualifying Test AFOQT

1. Find the distance between $(0,0)$ and $(3,4)$.

   $5$. Applies the distance formula, simplifying the square root of summed squares.

2. State the power of a power rule for $(a^m)^n$.

   $a^{mn}$. Raising a power to another power multiplies the exponents together.

3. What is the factoring identity for a difference of squares: $a^2-b^2$?

   $(a-b)(a+b)$. Recognizes the difference of squares pattern, factoring into conjugate binomials.

4. What is the exponent rule for multiplying powers with the same base: $a^m\cdot a^n$?

   $a^{m+n}$. When multiplying powers with the same base, add the exponents to combine them.

5. State the area formula for a circle with radius $r$.

   $A=\pi r^2$. Squares the radius and multiplies by pi to determine the enclosed area.

6. Find the midpoint of the segment from $(2,-1)$ to $(8,5)$.

   $\left(5,2\right)$. Averages the x-coordinates and y-coordinates separately using the midpoint formula.

7. What is the sum of the interior angles of a triangle in degrees?

   $180^\circ$. Derived from the fact that any triangle can be divided into two right triangles summing to 180 degrees.

8. State the point-slope form of a line with slope $m$ through $(x_1,y_1)$.

   $y-y_1=m(x-x_1)$. Expresses the line equation using a known point and slope to define its path.

9. State the distributive property for multiplying $a$ by $(b+c)$.

   $a(b+c)=ab+ac$. Multiplication distributes over addition, expanding the product into a sum of terms.

10. What is the midpoint formula for the segment joining $(x_1,y_1)$ and $(x_2,y_2)$?

    $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Averages the coordinates to locate the central point of the segment.

11. Factor completely: $x^2-9$.

    $(x-3)(x+3)$. Applies the difference of squares formula, where 9 is 3 squared.

12. State the circumference formula for a circle with radius $r$.

    $C=2\pi r$. Multiplies the diameter by pi to find the perimeter of the circle.

13. What is the factoring identity for a perfect square trinomial: $a^2+2ab+b^2$?

    $(a+b)^2$. Identifies the square of a binomial, where the middle term is twice the product of the roots.

14. Find the area of a triangle with base $10$ and height $6$.

    $30$. Multiplies base by height and divides by 2 according to the triangle area formula.

15. State the area formula for a triangle with base $b$ and height $h$.

    $A=\frac{1}{2}bh$. Computes the space enclosed by multiplying base by height and halving for the triangular shape.

16. What is the exponent rule for dividing powers with the same base: $\frac{a^m}{a^n}$, $a\ne 0$?

    $a^{m-n}$. When dividing powers with the same base, subtract the exponents to simplify the quotient.

17. State the quadratic formula for solutions to $ax^2+bx+c=0$, $a\ne 0$.

    $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived from completing the square, it solves quadratic equations by isolating the variable.

18. Solve for $x$: $x^2-5x+6=0$.

    $x=2$ or $x=3$. Factors into binomials whose product is 6 and sum is -5, then sets each to zero.

19. Find the sum of the interior angles of a pentagon in degrees.

    $540^\circ$. Uses the formula (n-2)*180 degrees for an n-sided polygon, where n=5 for a pentagon.

20. State the distance formula between points $(x_1,y_1)$ and $(x_2,y_2)$.

    $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Applies the Pythagorean theorem in two dimensions to find the straight-line distance.

21. Find the slope of the line through $(2,3)$ and $(6,11)$.

    $m=2$. Uses the slope formula with the change in y over change in x between the points.

22. What is the slope formula between points $(x_1,y_1)$ and $(x_2,y_2)$?

    $m=\frac{y_2-y_1}{x_2-x_1}$. Calculates the rate of change as the ratio of vertical to horizontal differences between points.

23. State the Pythagorean theorem for a right triangle with legs $a,b$ and hypotenuse $c$.

    $a^2+b^2=c^2$. Relates the sides of a right triangle, where the square of the hypotenuse equals the sum of squares of legs.

24. Simplify the expression $2x-3(x-4)$.

    $12-x$. Distribute the -3, combine like terms to simplify the expression.

25. Find $x$ if $3(x-2)=15$.

    $x=7$. Distribute the 3, add 6 to both sides, then divide by 3 to isolate x.

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