Question

1. partial point is located at xy coordinates: a) 3,-2 b)-3,2 c)-3,-2 d)2,-3 fill in the correct answer: (a b c d) with coordinate grid image
2. two right triangles, one with leg 4 and 12, the other with leg 6 and q side q must be equal to: a) 24 b)9 c)14 d)18 fill in the correct answer: (a b c d) with triangle diagrams
3. rectangular prism with 6 in, 15 in, 8 in paula decides to fill her aquarium 1/5 full of water. that would give her how many cubic inches of water? a) 5/48 b) 104 c) 144 d) 90 fill in the correct answer: (a b c d) with prism diagram
4. triangle with 100 degrees and 40 degrees, angle m how many degrees is angle m? a) 180 b) 40 c) 140 d) 40 partial fill in the correct answer: (a b c d) with triangle diagram
5. the little box is 1 ft on each side. how many little boxes will fit into the large box? large box 6 ft on each side a) 6 b) 66 c) 216 d) 36 fill in the correct answer: (a b c d) with cube diagrams

and: geometry and spatial sense number correct on this

Explanation:

Question 12

Step1: Identify similar triangles

The two triangles are right - angled and similar (since they have the same shape, right - angled). For similar triangles, the ratios of corresponding sides are equal. Let the sides of the first triangle be \(a = 4\), \(b=12\) and the sides of the second triangle be \(a'=6\), \(b' = Q\). The ratio of the corresponding sides should be equal, so \(\frac{4}{6}=\frac{12}{Q}\) (or we can also think of it as the ratio of the smaller side to the larger side of the same - type in each triangle). Another way: The ratio of the base of the first triangle to the base of the second triangle is \(\frac{4}{6}=\frac{2}{3}\), so the ratio of the height (the other leg) should also be \(\frac{2}{3}\). Let the height of the second triangle be \(Q\), then \(\frac{12}{Q}=\frac{2}{3}\), cross - multiply: \(2Q = 12\times3=36\), so \(Q = 18\)? Wait, no, wait. Wait, maybe I mixed up the sides. Wait, the first triangle has base 4 and height 12, the second has base 6 and height \(Q\). Since they are similar, \(\frac{4}{6}=\frac{12}{Q}\)? No, that would be wrong. Wait, actually, the ratio of corresponding sides: if the first triangle's base is 4 and the second's base is 6, then the scale factor is \(\frac{6}{4}=\frac{3}{2}\). So the height of the second triangle should be \(12\times\frac{3}{2}=18\).

Step2: Calculate the value of Q

Using the property of similar triangles (ratio of corresponding sides is equal). Let the two right - angled triangles be \(\triangle ABC\) (with \(AB = 4\), \(BC = 12\)) and \(\triangle DEF\) (with \(DE=6\), \(EF = Q\)). Since \(\triangle ABC\sim\triangle DEF\), \(\frac{AB}{DE}=\frac{BC}{EF}\). Substituting the values: \(\frac{4}{6}=\frac{12}{Q}\)? No, wait, that is incorrect. Wait, actually, the ratio of the sides of the first triangle to the second triangle: if the first triangle has sides 4 (base) and 12 (height), the second has 6 (base) and \(Q\) (height). The scale factor from the first to the second is \(\frac{6}{4}=\frac{3}{2}\). So the height of the second triangle \(Q=12\times\frac{3}{2} = 18\).

Step1: Calculate the volume of the aquarium

The aquarium is a rectangular prism. The formula for the volume of a rectangular prism is \(V=l\times w\times h\), where \(l = 15\) in, \(w = 6\) in, \(h = 8\) in. So \(V=15\times6\times8\). Calculate \(15\times6 = 90\), then \(90\times8=720\) cubic inches.

Step2: Calculate the volume of water

Paula fills it \(\frac{1}{5}\) full. So the volume of water \(V_{water}=\frac{1}{5}\times720\). \(\frac{720}{5}=144\) cubic inches.

Step1: Recall the triangle angle sum property

The sum of the interior angles of a triangle is \(180^{\circ}\). Let the three angles of the triangle be \(A = 100^{\circ}\), \(B = 40^{\circ}\) and \(C=m\). Then \(A + B + C=180^{\circ}\).

Step2: Solve for angle m

Substitute the known values: \(100^{\circ}+40^{\circ}+m = 180^{\circ}\). Combine like terms: \(140^{\circ}+m = 180^{\circ}\). Subtract \(140^{\circ}\) from both sides: \(m=180^{\circ}- 140^{\circ}=40^{\circ}\)? Wait, no, wait. Wait, the triangle has angles 100 degrees, 40 degrees, and \(m\). Wait, \(100 + 40+m=180\), so \(m = 180-(100 + 40)=40\)? But the options have b) 40 and d) 40? Wait, maybe the triangle is an isosceles? Wait, no, the sum of angles in a triangle is 180. So \(100 + 40+m=180\), so \(m = 40\). So the answer is b) 40.

Answer:

d) 18

Question 13