In isosceles triangle DEF with DE cong DF, each base angle measures 58°. Which side has the greatest length?

DE and DF are equal and longest

Geometric reasoning Practice Questions (Grade 9)

Warm-upeasy

In isosceles $\triangle DEF$ with $DE \cong DF$ , each base angle measures $58^{\circ}$ . Which side has the greatest length?

The base angles at $E$ and $F$ each measure $58^{\circ}$ , so the vertex angle at $D$ is $180^{\circ} - 2(58^{\circ}) = 64^{\circ}$ . The greatest angle is at $D$ , so the opposite side $EF$ is the longest.

Mid-gameeasy

In the standard coordinate plane, how many points are both $5$ units from the origin and also exactly $3$ units from the line $x = 0$ ?

Points $5$ units from the origin satisfy $x^{2} + y^{2} = 25$ . Being exactly $3$ units from $x = 0$ means $|x| = 3$ , so $x = 3$ or $x = -3$ . Substituting gives $y^{2} = 16$ , so $y = 4$ or $y = -4$ for each $x$ -value. That yields $(3, 4)$ , $(3, -4)$ , $(-3, 4)$ , and $(-3, -4)$ : $4$ points.

Mid-gameeasy

Parallel lines $m$ and $n$ are cut by a transversal. At one intersection, an angle measures $115^{\circ}$ . What is the measure of the corresponding angle at the other intersection?

When parallel lines are cut by a transversal, corresponding angles are congruent. The corresponding angle at the second intersection also measures $115^{\circ}$ .

Mid-gameeasy

In $\triangle ABC$ , the exterior angle at vertex $C$ measures $130^{\circ}$ . If $m\angle A = 48^{\circ}$ , what is $m\angle B$ ?

An exterior angle equals the sum of the two remote interior angles: $m\angle A + m\angle B = 130^{\circ}$ . So $m\angle B = 130^{\circ} - 48^{\circ} = 82^{\circ}$ .

Mid-gameeasy

Each exterior angle of a regular polygon measures $30^{\circ}$ . How many sides does the polygon have?

The exterior angles of any convex polygon sum to $360^{\circ}$ . For a regular polygon, $n \cdot 30^{\circ} = 360^{\circ}$ , so $n = 12$ .

Mid-gameeasy

Two similar triangles have perimeters in the ratio $3 : 5$ . The shortest side of the smaller triangle is $6$ . What is the length of the shortest side of the larger triangle if corresponding sides are compared?

Similar figures have a constant scale factor between corresponding lengths. Perimeter ratio $3 : 5$ means each side of the larger triangle is $\dfrac{5}{3}$ times the matching side of the smaller. The larger shortest side is $6 \cdot \dfrac{5}{3} = 10$ .

Buzzer beatermedium

In the standard coordinate plane, how many points with integer coordinates lie on the circle $x^{2} + y^{2} = 25$ ?

Integer solutions satisfy $a^{2} + b^{2} = 25$ . On the axes: $(\pm 5, 0)$ and $(0, \pm 5)$ give $4$ points. Off the axes: $(\pm 3, \pm 4)$ and $(\pm 4, \pm 3)$ each contribute $4$ sign pairs, for $8$ more. The total is $4 + 8 = 12$ points.

Geometric reasoning. Game on.

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