HSPT Math : Concepts

Study concepts, example questions & explanations for HSPT Math

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Example Questions

Example Question #1471 : Concepts

What is the total degrees of the angles in a square?

Possible Answers:

\(\displaystyle 360\)

\(\displaystyle 90\)

\(\displaystyle 270\)

\(\displaystyle 180\)

Correct answer:

\(\displaystyle 360\)

Explanation:

A square has four right angles which each have \(\displaystyle 90\) degrees.  

To get the total you just multiple the measure of one by \(\displaystyle 4\) to get 

\(\displaystyle 90*4=360\).

Example Question #11 : Plane Geometry

Angle \(\displaystyle \angle ABC\) measures \(\displaystyle 20^{\circ}\)

 \(\displaystyle \overrightarrow{BD}\) is the bisector of \(\displaystyle \angle ABC\)

 \(\displaystyle \overrightarrow{BE}\) is the bisector of \(\displaystyle \angle CBD\)

What is the measure of \(\displaystyle \angle ABE\)?

Possible Answers:

\(\displaystyle 10^{\circ}\)

\(\displaystyle 40^{\circ}\)

\(\displaystyle 15^{\circ}\)

\(\displaystyle 30^{\circ}\)

\(\displaystyle 5^{\circ}\)

Correct answer:

\(\displaystyle 15^{\circ}\)

Explanation:

Angle pic

Let's begin by observing the larger angle. \(\displaystyle \angle ABC\) is cut into two 10-degree angles by \(\displaystyle \overrightarrow{BD}\). This means that angles \(\displaystyle \angle ABD\) and \(\displaystyle \angle CBD\) equal 10 degrees. Next, we are told that \(\displaystyle \overrightarrow{BE}\) bisects \(\displaystyle \angle CBD\), which creates two 5-degree angles.  \(\displaystyle \angle ABE\) consists of \(\displaystyle \angle ABD\), which is 10 degrees, and \(\displaystyle \angle DBE\), which is 5 degrees. We need to add the two angles together to solve the problem.

\(\displaystyle \angle ABE=\angle ABD+\angle DBE\)

\(\displaystyle \angle ABE=10^{\circ}+5^{\circ}\)

\(\displaystyle \angle ABE=15^{\circ}\)

Example Question #1472 : Concepts

If  \(\displaystyle \overleftrightarrow{FH} \parallel \overleftrightarrow{AD}\), \(\displaystyle \overline{BE}\perp \overleftrightarrow{GC}\), and \(\displaystyle m\angle HGC=58^\circ\), what is the measure, in degrees, of \(\displaystyle \angle ABE\)

Alternate interior angles   

 

Possible Answers:

122

62

148

58

32

Correct answer:

148

Explanation:

The question states that \(\displaystyle \overleftrightarrow{FH} \parallel \overleftrightarrow{AD}\). The alternate interior angle theorem states that if two parallel lines are cut by a transversal, then pairs of alternate interior angles are congruent; therefore, we know the following measure:

\(\displaystyle m\angle GCA = 58^\circ\)

The sum of angles of a triangle is equal to 180 degrees. The question states that \(\displaystyle \overline{BE}\perp \overleftrightarrow{GC}\); therefore we know the following measure:

\(\displaystyle m \angle BEC = 90^\circ\)

Use this information to solve for the missing angle: \(\displaystyle \angle EBC\)

\(\displaystyle 180^\circ=m\angle EBC+58^\circ+90^\circ\)

\(\displaystyle m\angle EBC=32^\circ\)

The degree measure of a straight line is 180 degrees; therefore, we can write the following equation:

\(\displaystyle 180^\circ=m\angle ABE+32^\circ\)

\(\displaystyle m\angle ABE=148^\circ\)

The measure of \(\displaystyle \angle ABE\) is 148 degrees. 

Example Question #1472 : Concepts

What is the slope of a line through the points \(\displaystyle (2,5)\) and \(\displaystyle (-2,3)\) ?

Possible Answers:

\(\displaystyle \frac{1}{2}\)

Undefined slope

\(\displaystyle -\frac{1}{2}\)

\(\displaystyle -2\)

\(\displaystyle 2\)

Correct answer:

\(\displaystyle \frac{1}{2}\)

Explanation:

Use the slope fomula, setting \(\displaystyle x_{1} = -2, x_{2} = 2,y_{1} = 3, y_{2} = 5\)

\(\displaystyle m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{5-3}{2-(-2)} = \frac{2}{4} =\frac{1}{2}\)

Example Question #1474 : Concepts

In which quadrant, or on which axis, is the point with coordinates \(\displaystyle (6,-6)\) located?

Possible Answers:

Quadrant II

The \(\displaystyle x\)-axis

Quadrant III

Quadrant IV

Quadrant I

Correct answer:

Quadrant IV

Explanation:

Any point with a positive \(\displaystyle x\)-coordinate and a negative \(\displaystyle y\)-coordinate is located in the lower right quadrant - Quadrant IV.

Example Question #292 : Geometry

Which of the following is an equation of a line with slope 0?

Possible Answers:

\(\displaystyle y = x+7\)

\(\displaystyle y=x^{2} +1\)

\(\displaystyle x+y= 10\)

\(\displaystyle y = -5\)

\(\displaystyle x = 6\)

Correct answer:

\(\displaystyle y = -5\)

Explanation:

A line with slope 0 has equation \(\displaystyle y = b\) for some real value \(\displaystyle b\).

Example Question #293 : Geometry

Axes

 

Give the equation of the red line in slope-intercept form.

Possible Answers:

\(\displaystyle y =- \frac{7}{4}x + 6\)

\(\displaystyle y = \frac{7}{4}x + 6\)

\(\displaystyle y = -\frac{4}{7}x + 6\)

\(\displaystyle y =- \frac{7}{4}\left (x + 6 \right )\)

\(\displaystyle y = \frac{4}{7}x + 6\)

Correct answer:

\(\displaystyle y = \frac{7}{4}x + 6\)

Explanation:

The slope of the line is 

\(\displaystyle m = \frac{y_{2}- y_{1}}{x_{2}- x_{1}} = \frac{6- (-1)}{0-(-4)} = \frac{7}{4}\)

The \(\displaystyle y\)-intercept of the line has \(\displaystyle y\)-coordinate \(\displaystyle b= 6\)

The slope-intercept form can be written:

\(\displaystyle y = mx+b\)

Replace:

\(\displaystyle y = \frac{7}{4}x+6\)

Example Question #294 : Geometry

Axes_2

 

The green and blue lines are perpendicular: What is the slope of the blue line?

Possible Answers:

\(\displaystyle \frac{1}{3}\)

\(\displaystyle -3\)

\(\displaystyle -\frac{1}{3}\)

It cannot be determined from the information given.

\(\displaystyle 3\)

Correct answer:

\(\displaystyle 3\)

Explanation:

The slope of the blue line, being perpendicular to the green line, is the opposite of the reciprocal of the slope of the green line. The slope of the green line can be found using the slope formula:

\(\displaystyle m = \frac{y_{2}- y_{1}}{x_{2}- x_{1}} = \frac{10-0}{0-30} = -\frac{1}{3}\)

The opposite of the reciprocal of \(\displaystyle -\frac{1}{3}\) is 3, and this is the slope of the blue line.

Example Question #1 : How To Find The Midpoint Of A Line Segment

A line segment on the coordinate plane has endpoints \(\displaystyle (a-5,-b)\) and \(\displaystyle (b+8, a-4)\). In terms of \(\displaystyle a\) and \(\displaystyle b\), as applicable, give the \(\displaystyle x\)-coordinate of its midpoint.

Possible Answers:

\(\displaystyle a + b + 3\)

\(\displaystyle \frac{a + b + 3}{2}\)

\(\displaystyle \frac{a - b - 4}{2}\)

\(\displaystyle \frac{a - b - 13}{2}\)

\(\displaystyle a - b - 4\)

Correct answer:

\(\displaystyle \frac{a + b + 3}{2}\)

Explanation:

The \(\displaystyle x\)-coordinate of the midpoint of a line segment is the mean of the \(\displaystyle x\)-coordinates of its endpoints. Therefore, the \(\displaystyle x\)-coordinate is 

\(\displaystyle x= \frac{(a-5)+(b+8)}{2} = \frac{a+b-5+8}{2} = \frac{a+b+3}{2}\).

Example Question #1475 : Concepts

A line segment on the coordinate plane has endpoints \(\displaystyle (a-5,-b)\) and \(\displaystyle (b+8, a-4)\). In terms of \(\displaystyle a\) and \(\displaystyle b\), as applicable, give the \(\displaystyle y\)-coordinate of its midpoint.

Possible Answers:

\(\displaystyle \frac{a + b - 4}{2}\)

\(\displaystyle \frac{a + b + 3}{2}\)

\(\displaystyle a + b + 3\)

\(\displaystyle a - b - 4\)

\(\displaystyle \frac{a - b - 4}{2}\)

Correct answer:

\(\displaystyle \frac{a - b - 4}{2}\)

Explanation:

The \(\displaystyle y\)-coordinate of the midpoint of a line segment is the mean of the \(\displaystyle y\)-coordinates of its endpoints. Therefore, the \(\displaystyle y\)-coordinate is 

\(\displaystyle y= \frac{(a-4)+(-b)}{2} = \frac{a-b-4}{2}\).

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