First, multiple choice questions

The lengths of the three sides of the 1. triangle are 6,8 and 10 respectively, so the height of its shortest side is ................... ().

A.4 B. 5 C. 6 D. 8

2. The square of the length of each side of a triangle (from small to large) is as follows, among which those that are not right-angled triangles are

A. 1: 1:2 b . 1:3:4 c . 9:25:36d . 25: 144: 169

* 3. Let the two right-angled sides of a right-angled triangle be A and B, the height of the hypotenuse be H, and the length of the hypotenuse be C, then the shape of the triangle with side lengths of c+h, a+b and H is ....................................... ().

A. right triangle B. acute triangle C. obtuse triangle D. uncertainty

* 4. In △ ABC, ∠A:∠B:∠C= 1:2:3, then BC:AC:AB is .......................... ().

A. 1:2:3 b . 1:2:c . 1::2d . 1:2

5. In △ ABC, AB= 15, AC= 13. High AD= 12. Then the circumference of △ABC is ................ ().

A.42 B. 32 C. 42 or 32 D. 37 or 33

Tip: There are two situations.

Second, fill in the blanks

1. If there are two line segments with lengths of 8 cm and 17cm respectively, and the length of the third line segment satisfies the condition of _ _ _ _ _ _ _ _, then these three line segments can form a right triangle.

2. The carpenter made a rectangular desktop with a length of 60cm, a width of 32cm and a diagonal length of 68cm. This desktop is _ _ _ _ _ _ (fill in "qualified" or "unqualified").

3. As shown in the figure, there is a cylinder with a height of 12cm and a bottom radius.

It is 3cm, and there is an ant at the bottom A of the cylinder. It wants the top.

B Food, the shortest distance ants pass is _ _ _ _ _ _ _ cm. (π takes 3)

4. As shown in the figure, there is a right-angled triangular paper with two right-angled sides AC=6cm and BC=8cm. Now fold the right-angled side AC along the straight line AD, so that it falls on the hypotenuse AB, which coincides with AE, and CD is equal to _ _ _ _ _ _ _ _.

Third, the calculation problem

1. As shown in the figure, MN Highway and PQ Highway meet at point P, there is a middle school at point A, AP = 160m, and the distance from point A to MN Highway is 80m. If the tractor will be affected by the noise in the range of 100m when driving, will the school be affected when the tractor is driving along the PN direction on the MN expressway? Please explain the reason. If the speed of the tractor is known as 18km/h, how long will the school be affected?

2. It is known that the lengths of the three sides of a right triangle are 3, 4 and x respectively, so find x2.

In the summer vacation, Xiao Ming went to an island to explore treasures. As shown in the picture, he boarded the island.

Go 8 kilometers east, 2 kilometers north, and then go west after encountering obstacles.

Go three kilometers, then turn six kilometers north, and then turn east. Only 1 km will find the treasure spot.

Treasure, what is the straight line distance from the landing point to the treasure point?

Landing point

There is a 2.5-meter-long ladder leaning against the vertical wall, and the heel of the ladder is 0.7 meters away from the bottom of the wall. If the top of the ladder drops 0.4 meters, how many meters is it from the root of the ladder to the bottom of the wall?

5. As shown in the figure, AB is a big tree with two monkeys at D, which is 10 meters away from the ground. At the same time, they found a basket of fruit at C. A monkey climbed from D to A at the top of the tree, then slid to C along the sliding rope AC, and another monkey slipped from D to B. When running from B to C, it was known that the distance traveled by the two monkeys was 15 meters, so the height AB of the tree could be calculated.

6. If the three sides A, B and C of △ABC satisfy A2+B2+C2+338 =10A+24b+26c, is △ABC a right triangle? Why?

* 7. In △ABC, BC= 1997, AC= 1998, AB2= 1997+ 1998, is △ABC a right triangle? Why?

8. In a square ABCD, e is the midpoint of BC, f is a point on CD, and CF= CD. Try to judge whether △AEF is a right triangle? Try to explain why.

9. An ant is at the vertex A of a rectangle, and a fly is at the vertex opposite to the spider on this rectangle.

At C 1, as shown in the figure, it is known that the rectangle is 6cm long, 5 cm wide and 3 cm high. Spiders are eager to catch flies along the rectangle.

In order to climb the surface of a spider, it must climb from point A to point C 1. There are many routes, long and short. What should spiders follow?

What is the shortest route to climb up? Can you help the spider find the shortest distance?

10. Enlarge three sides of a right triangle by the same multiple. Triangle or right triangle? What? Your reasons.

* 1 1. There is a cylindrical oil tank with a bottom circumference of12m and a height of 5m. You need to use point A to build a ladder around the oil tank, just above point A. How many meters is the shortest ladder?

* 12. The length, width and height of the wooden box are 40dm, 30dm and 50dm respectively, and there is also a 70dm wooden stick. Can you put it in? Please provide a justification for the answer.

13. Given three sides A, B and C of △ABC, and a+b= 17, ab=60, C = 13, is △ ABC a right triangle? Can you explain why?

14. As shown in the figure, two stations A and B (regarded as two points on a straight line) are 25km apart, and two villages C and D (regarded as two points).

DA⊥AB is located in A and CB⊥AB is located in B. It is known that DA= 15km and CB= 10km. Now it is necessary to build a local product acquisition point on the railway.

Station E, so that the distance between villages C and D and station E is equal, how far is station E from station A?

* 15. Known: As shown in the figure, in △ABC, ∠ c = 90,

∠ 1=∠2, CD= 1.5, BD=2.5, and find the length of AC.

16. Known: As shown in the figure, find the length of MN at △ABC, ∠ ACB = 90, AC= 12, CB=5, AM=AC, BN=BC.

* 17. kudzu vine is a difficult plant. Its own waist is not hard. In order to compete for rain and sunshine, it often wanders around the trunk. It also has a unique skill, that is, its route around the tree always follows the short route-circling forward. Do plants know math?

If you read the above information, can you design a method to solve the following problems?

(1) If the circumference of a tree is 3cm, it rises 4cm in a circle, how many centimeters does it crawl?

(2) If a tree is 8 cm in circumference and it crawls around 10 cm, how many centimeters will it rise when it crawls around? If you climb 10 times to reach the top of the tree, how many centimeters is the trunk high?

18. As shown in the figure, e is the midpoint of the side CD of the square ABCD. Extending AB to F makes BF= AB, then does FE equal FA? Why?

* 19. As shown in the figure, ∠ A = 60, ∠ B = ∠ D = 90. If BC=4 and CD=6, find the length of AB.

* 20. As shown in the figure, ∠xoy = 60, m is a point in ∠xoy, and its distance to ox is 2. Its distance to oy is 1 1. Find the length of OM.