Quadrilateral 2

Show that the quadrilateral with vertices P1(0,1), P2(4,2) P3(3,6) P4(-5,4) has two right triangles.

Result

x =  2

Solution:

x1=0 y1=1 x2=4 y2=2 x3=3 y3=6 x4=−5 y4=4 a=(x1−x2)2+(y1−y2)2=(0−4)2+(1−2)2≐17≐4.1231 b=(x2−x3)2+(y2−y3)2=(4−3)2+(2−6)2≐17≐4.1231 c=(x3−x4)2+(y3−y4)2=(3−(−5))2+(6−4)2≐2 17≐8.2462 d=(x4−x1)2+(y4−y1)2=((−5)−0)2+(4−1)2≐34≐5.831 u1=(x1−x3)2+(y1−y3)2=(0−3)2+(1−6)2≐34≐5.831 u2=(x2−x4)2+(y2−y4)2=(4−(−5))2+(2−4)2≐85≐9.2195 t1=u12−a2−b2=5.8312−4.12312−4.12312=0 t2=u22−b2−c2=9.21952−4.12312−8.24622=−0 t3=u12−c2−d2=5.8312−8.24622−5.8312=−68 t4=u22−d2−a2=9.21952−5.8312−4.12312=34 t1=0=>P1P2P3=90∘ t2=0=>P2P3P4=90∘ x=2 angles=2x_{1}=0 \ \\ y_{1}=1 \ \\ x_{2}=4 \ \\ y_{2}=2 \ \\ x_{3}=3 \ \\ y_{3}=6 \ \\ x_{4}=-5 \ \\ y_{4}=4 \ \\ a=\sqrt{ (x_{1}-x_{2})^2+(y_{1}-y_{2})^2 }=\sqrt{ (0-4)^2+(1-2)^2 } \doteq \sqrt{ 17 } \doteq 4.1231 \ \\ b=\sqrt{ (x_{2}-x_{3})^2+(y_{2}-y_{3})^2 }=\sqrt{ (4-3)^2+(2-6)^2 } \doteq \sqrt{ 17 } \doteq 4.1231 \ \\ c=\sqrt{ (x_{3}-x_{4})^2+(y_{3}-y_{4})^2 }=\sqrt{ (3-(-5))^2+(6-4)^2 } \doteq 2 \ \sqrt{ 17 } \doteq 8.2462 \ \\ d=\sqrt{ (x_{4}-x_{1})^2+(y_{4}-y_{1})^2 }=\sqrt{ ((-5)-0)^2+(4-1)^2 } \doteq \sqrt{ 34 } \doteq 5.831 \ \\ u_{1}=\sqrt{ (x_{1}-x_{3})^2+(y_{1}-y_{3})^2 }=\sqrt{ (0-3)^2+(1-6)^2 } \doteq \sqrt{ 34 } \doteq 5.831 \ \\ u_{2}=\sqrt{ (x_{2}-x_{4})^2+(y_{2}-y_{4})^2 }=\sqrt{ (4-(-5))^2+(2-4)^2 } \doteq \sqrt{ 85 } \doteq 9.2195 \ \\ t_{1}=u_{1}^2 - a^2-b^2=5.831^2 - 4.1231^2-4.1231^2=0 \ \\ t_{2}=u_{2}^2 - b^2-c^2=9.2195^2 - 4.1231^2-8.2462^2=-0 \ \\ t_{3}=u_{1}^2 - c^2-d^2=5.831^2 - 8.2462^2-5.831^2=-68 \ \\ t_{4}=u_{2}^2 - d^2-a^2=9.2195^2 - 5.831^2-4.1231^2=34 \ \\ t_{1}=0=> P_{1}P_{2}P_{3}=90^\circ \ \\ t_{2}=0=> P_{2}P_{3}P_{4}=90^\circ \ \\ x=2 \ \text{angles}=2

Try calculation via our triangle calculator.


Try another example


Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!





Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Showing 0 comments:
1st comment
Be the first to comment!
avatar




Tips to related online calculators
For Basic calculations in analytic geometry is helpful line slope calculator. From coordinates of two points in the plane it calculate slope, normal and parametric line equation(s), slope, directional angle, direction vector, the length of segment, intersections the coordinate axes etc.
Do you want to convert length units?
Pythagorean theorem is the base for the right triangle calculator.
See also our trigonometric triangle calculator.

You need to know the following knowledge to solve this word math problem:


 

 

 

Next similar math problems:

  1. Vertices of RT
    RightTriangleMidpoint_3 Show that the points P1 (5,0), P2 (2,1) & P3 (4,7) are the vertices of a right triangle.
  2. Spruce height
    stromcek_7 How tall was spruce that was cut at an altitude of 8m above the ground and the top landed at a distance of 15m from the heel of the tree?
  3. Trapezium ABCD
    lichobeznik_5 In the figure, ABDC is a trapezium in which AB || CD. line segments RN and LM are drawn parallel to AB such that AJ=JK=KP. If AB=0.5m and AP=BQ=1.8m, find the lengths of AC, BD, RN and LM. angle D=angle C=60
  4. Triangle IRT
    triangles_5 In isosceles right triangle ABC with right angle at vertex C is coordinates: A (-1, 2); C (-5, -2) Calculate the length of segment AB.
  5. The ditch
    prikop Ditch with cross section of an isosceles trapezoid with bases 2m 6m are deep 1.5m. How long is the slope of the ditch?
  6. Distance
    origin_math Wha is the distance between the origin and the point (18; 22)?
  7. Diamond
    cerveny_kosoctverec Side length of diamond is 35 cm and the length of the diagonal is 56 cm. Calculate the height and length of the second diagonal.
  8. Park
    diamond_3 In the park is marked diamond shaped line connecting locations A, D, S, C, B, A. Calculate its length if |AB| = 108 m, |AC| = 172.8 m.
  9. Satin
    diagonal_rectangle_3 Sanusha buys a piece of satin 2.4 m wide. The diagonal length of the fabric is 4m. What is the length of the piece of satin?
  10. Diagonals in diamons/rhombus
    diamonds Rhombus ABCD has side length AB = 4 cm and a length of one diagonal of 6.4 cm. Calculate the length of the other diagonal.
  11. Right 24
    euclid_theorem Right isosceles triangle has an altitude x drawn from the right angle to the hypotenuse dividing it into 2 unequal segments. The length of one segment is 5 cm. What is the area of the triangle? Thank you.
  12. Median
    1taznice_rt In the right triangle are sides a=41 dm b=42 dm. Calculate the length of the medians tc to the hypotenuse.
  13. Tv screen
    tv2 The size of a tv screen is given by the length of its diagonal. If the dimension of a tv screen is 16 inches by 14 inches, what is the size of the tv screen?
  14. Distance between 2 points
    axes2 Find the distance between the points (7, -9), (-1, -9)
  15. KLMN
    trapezium In the trapezoid KLMN is given this informations: 1. segments KL and MN are parallel 2. segments KL and KM has same length 3. segments KN, NM and ML has same length. Determine the size of the angle KMN.
  16. Reference angle
    anglemeter Find the reference angle of each angle:
  17. Theorem prove
    thales_1 We want to prove the sentence: If the natural number n is divisible by six, then n is divisible by three. From what assumption we started?